Rising inequality: transitory or persistent? New evidence from a panel of U.S. tax returns.

Debacker, Jason ; Heim, Bradley ; Panousi, Vasia 等

VII. The Role of the Federal Tax System

This section explores the role of the federal tax system in the increase in income inequality over our sample period. In particular, we examine whether the trend in inequality for after-tax household income differs materially from that for pre-tax income. As discussed in section II.B, our measure of after-tax household income reflects all federal personal income taxes (obtained from Form 1040), including all refundable tax credits such as the earned income tax credit and the child tax credit, as well as payroll taxes (calculated using information from W-2 forms).

The last two columns of table 4 present point estimates and standard errors for our ECM estimated on after-tax household income using both our male-headed households sample and our broader sample of all households. Figure 9 plots the total, persistent, and transitory variances of both pre-tax and after-tax household income for the sample of all households. As the figure shows, the total variance of after-tax income is on average 0.10 squared log point, or roughly 15 percent, smaller than the variance of pre-tax income, reflecting the overall progressivity of the federal tax system. The effect of the tax system in reducing income inequality appears relatively stable over the sample period, but for the period as a whole, pretax household income inequality increased by more than after-tax income inequality (0.13 versus 0.08 squared log point). That is, the tax system appears to have reduced the increase in household income inequality over the sample period. Nonetheless, as was already seen in figure 1, this attenuating effect was insufficient to alter the broad trend toward rising inequality for after-tax household income.

[FIGURE 9 OMITTED]

The relatively constant effect of the federal tax system on reducing the level of inequality during our sample period might appear surprising in light of the high-profile reductions in marginal tax rates, especially at the high end of the income distribution, in 2001 and 2003. However, the changes in top marginal tax rates were accompanied by (smaller) reductions in marginal tax rates for other income groups as well as by significant expansions of the earned income tax credit and the child tax credit. Our results suggest that the net effect on after-tax income inequality of all these changes to the federal tax system was relatively small. (46)

VIII. Conclusions

We have used a confidential panel of tax returns from the Internal Revenue Service to analyze the role of persistent and transitory income components in changes in inequality in male labor earnings and total household income, both before and after taxes, in the United States over the period 1987-2009. We first documented an increase in inequality in male earnings and in

pre-tax and after-tax household income in our data during this period, consistent with what other studies have documented using different data sets. We then examined the contributions of persistent and transitory income components to this increase in inequality, as measured by the cross-sectional variance of log income.

We have used two broad sets of methods in our analysis. First, we employed a variety of simple nonparametric decomposition methods that use a strict definition of transitory income, which is not allowed to be serially correlated, and a broad definition of persistent income, which captures income with varying degrees of persistence. Second, we employed rich nonstationary error components models of income dynamics, which fully specify the process that generates income over time, and essentially decomposed income into a highly persistent piece and another transitory piece that allows for some limited degree of serial correlation. Our paper is the first to estimate rich nonstationary ECMs of income on U.S. administrative data, and among the first to apply nonstationary ECMs to household-level income. Here the quality and significant size of our data set allow us to obtain very precise estimates of our models.

Overall, our data yield very robust results for the trends in the variance of persistent and transitory income components. For male labor earnings, we find that the variance of the persistent component of earnings increased over the sample period, but the variance of the transitory component did not. Hence the increase in male earnings inequality was driven entirely by the increase in the persistent component, thus reflecting an increase in persistent inequality. For household income, both before and after taxes, the increase in inequality over this period derived mostly (although not entirely) from the persistent component. The increase in the variance of the transitory component of total household income reflects an increase in the transitory variance of spousal labor earnings and investment income. We also find evidence that the federal tax system helped reduce the increase in household income inequality, but this attenuating effect was insufficient to significantly alter the broad trend toward rising inequality.

Our findings, along with economic theory, suggest that the increase in income inequality observed in roughly the last two decades should translate into increases in consumption inequality and is therefore likely to be welfare-reducing, at least according to most social welfare functions. Although measurement problems with household consumption data in the United States have made it difficult to convincingly measure the increase in consumption inequality, some recent studies that attempt to control for these measurement issues, such as Aguiar and Bils (2012) and Attanasio, Hurst, and Pistaferri (2012), suggest that it was indeed substantial. This is consistent with our findings of a large role of the persistent component of income in rising income inequality.

APPENDIX A

An Alternative Nonstationary ECM Specification

A few papers (for instance, Heathcote, Storesletten, and Violante 2010, Blundell, Pistaferri, and Preston 2008, and Heathcote, Perri, and Violante 2010) have estimated versions of an alternative nonstationary ECM specification, in which the variance of persistent shocks can change over calendar time, but which are simpler along other dimensions of the model. Here we present estimates for a version of this alternative specification in order to check the robustness of our results. The general model can be expressed as

(A.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

(A.2) [p.sup.i.sub.a,t] = [[psi][p.sup.i.sub.a-1,t-1] + [[phi].sub.t][[eta].sup.i.sub.a,t]

(A.3) [[tau].sup.i.sub.a,t] = [[pi].sub.t][[epsilon].sup.i.sub.a,t] + [[theta].sub.1][[pi].sub.t- 1][[epsilon].sup.i.sub.a-1,t-1] + [[theta].sub.2][[pi].sub.t-2][[epsilon].sup.i.sub.a-2,t-2]

(A.4) [[alpha].sup.i] ~ i.i.d.(0, [[sigma].sup.2.sub.[alpha]), [[eta].sup.i.sub.a,t] ~ i.i.d.(0, [[sigma].sup.2.sub.[eta]]), [[epsilon].sup.i.sub.a,t] ~ i.i.d.(0, [[sigma].sup.2.sub.[epsilon]]).

In this specification the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] parameter multiplies the [[alpha].sup.i] component only, and a new set of parameters [[phi].sub.t] allow the variance of the persistent shocks [[eta].sup.i.sub.a,t] to change over calendar time. (Note that parameters [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are different from the [[lambda].sub.t] in our baseline model, since [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] allow only the value of [[alpha].sup.i] to change over time, and not that of the persistent characteristics [p.sup.i.sub.a,t].) The previous studies typically use a simpler version of this model that excludes the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] from equation A. 1. For our purposes the inclusion of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] component is necessary, because we cannot remove the income variation that is due to characteristics such as education, and in our context it is key to allow the prices of such characteristics to change over time. (47)

Table A.1 presents point estimates and standard errors for the above model for male earnings and for total pre-tax household income, the latter using our sample of all households. Figure A. 1 shows the corresponding decompositions of the cross-sectional variance of male earnings. Note that the component of the variance labeled "persistent" is the sum of the contributions of both [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [p.sup.i.sub.a,t] to the cross- sectional variance. As in our baseline ECM, the persistent variance component displays a clearly increasing trend, rising from 0.39 squared log point in 1987 to 0.50 squared log point in 2009. Fitting a linear time trend to this series yields an estimated trend coefficient of 0.0041 (with a standard error of 0.0003), similar to that obtained with our baseline nonstationary specification. The transitory part of the variance, the lowest line in figure A. 1, again exhibits no trend (an estimated linear time trend yields a coefficient of essentially zero).

Figure A.2 separates the persistent variance component in this model into the contributions of the terms [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [p.sup.i.sub.a,t]. As the figure shows, the increase in this component is driven by an increase in the variance of [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], whereas the variance of [p.sup.i.sub.a,t] fluctuates but does not exhibit any clear trend. As the absence of a trend in var([p.sup.i.sub.a,t]) implies, the estimated variance of the persistent shocks ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) in table A.2 varies substantially from year to year but has remained relatively stable on average over our sample period. (48) For the question addressed in this paper, these results are very similar to those obtained with our baseline model.

[FIGURE A.1 OMITTED]

[FIGURE A.2 OMITTED]

[FIGURE A.3 OMITTED]

Figure A.3 shows the decomposition, using the alternative model, of the cross-sectional variance of total pre-tax household income for our sample of all households. Here, too, the results are similar to those obtained with our baseline specification. There is a clear rising trend in the persistent component of the variance, and this increase is concentrated in the first half of the sample period. The transitory variance component fluctuates but overall is largely flat, except perhaps for a small increase in the last few years of the period. Fitting a linear time trend to the persistent and transitory variance components yields trend coefficients of 0.0055 (0.0005) and 0.0005 (0.0005), respectively. Again, most of the increase in the cross-sectional variance of total pre-tax household income was driven by the variance of the persistent component of income. In fact, this specification implies that the transitory variance component played even less of a role than in our baseline model (compare the 0.0005 estimated trend coefficient on the transitory variance component with the 0.0013 coefficient shown in the bottom right panel of table 5).

[FIGURE A.4 OMITTED]

Figure A.4 shows the contributions of var([bar.[lambda]], [[alpha].sup.i]) and var([p.sup.i.sub.a,t]) to the persistent variance component in the same decomposition and indicates that, similar to the case of male earnings, the increase in the persistent variance component was driven by an increase in the variance of var([bar.[lambda]], [[alpha].sup.i]), that is, by an increase in the [bar.[lambda]]. Fitting a linear time trend to the var([p.sup.i.sub.a,t]) series yields a trend coefficient of 0.0008 (0.0005), implying only a minor increase of about 0.02 squared log point over 23 years.

Overall, for the question asked in this paper, the results obtained with this alternative specification are very similar to those obtained with our baseline model.

Table A.1. Estimates of the Alternative Nonstationary Error
Components Model (a)

 Male labor earnings

 Persistent Transitory
Parameter component component

[[sigma].sup.2.sub.[alpha]] 0.1458
 (0.0235)

[[??].sub.t] polynomial (b)
[b.sub.1] 0.0136
 (0.0419)
[b.sub.2] (x 10) 0.0190
 (0.0584)
[b.sub.3] (x 100) -0.0175
 (0.0337)
[b.sub.4] (x 1000) 0.0038
 (0.0069)
[psi] 0.9619
 (0.0058)
[[sigma].sup.2.sub.[eta]] 0.0296
 (0.0040)
[[theta].sub.1] 0.2396
 (0.0163)
[[theta].sub.2] 0.1353
 (0.0179)
[[sigma].sup.2.sub.[epsilon]] 0.1749
 (0.0166)

[phi] or [pi] (c)
1987 1.0000 1.0000
1988 1.1382 1.0867
 (0.3929) (0.0571)
1989 1.2149 1.0160
 (0.2613) (0.0584)
1990 0.8679 0.9985
 (0.3494) (0.0530)
1991 1.0022 0.9845
 (0.2808) (0.0536)
1992 0.7569 1.0887
 (0.3553) (0.0505)
1993 1.1759 1.0444
 (0.2280) (0.0483)
1994 0.0051 1.0659
 (0.2890) (0.0509)
1995 1.2536 1.0071
 (0.1737) (0.0580)
1996 0.7085 1.0319
 (0.3096) (0.0567)
1997 1.0256 0.9884
 (0.1961) (0.0552)
1998 0.7510 1.0236
 (0.2669) (0.0599)
1999 0.9245 1.0070
 (0.2239) (0.0578)
2000 0.8369 1.0463
 (0.2959) (0.0654)
2001 1.1827 0.9787
 (0.1766) (0.0613)
2002 1.2415 0.9859
 (0.1344) (0.0538)
2003 0.7567 1.0239
 (0.1758) (0.0601)
2004 1.0366 0.9898
 (0.1428) (0.0534)
2005 0.7861 1.0202
 (0.2186) (0.0565)
2006 1.0178 1.0685
 (0.1622) (0.0582)
2007 0.7559 1.0508
 (0.2257) (0.0553)
2008 1.2584 1.0277
 (0.1457) (0.0544)
2009 1.0208
 (0.0557)

 Pre-tax household
 income, all households

 Persistent Transitory
Parameter component component

[[sigma].sup.2.sub.[alpha]] 0.1313
 (0.0187)

[[??].sub.t] polynomial (b)
[b.sub.1] 0.0170
 (0.0425)
[b.sub.2] (x 10) 0.0488
 (0.0567)
[b.sub.3] (x 100) -0.0443
 (0.0316)
[b.sub.4] (x 1000) 0.0097
 (0.0064)
[psi] 0.9693
 (0.0041)
[[sigma].sup.2.sub.[eta]] 0.0248
 (0.0025)
[[theta].sub.1] 0.2877
 (0.0114)
[[theta].sub.2] 0.1703
 (0.0142)
[[sigma].sup.2.sub.[epsilon]] 0.1533
 (0.0122)

[phi] or [pi] (c)
1987 1.0000 1.0000
1988 1.3067 1.0068
 (0.2610) (0.0496)
1989 0.9528 1.0198
 (0.3390) (0.0443)
1990 0.9199 0.9910
 (0.3456) (0.0465)
1991 1.0393 0.9619
 (0.2798) (0.0418)
1992 0.0028 1.0833
 (0.3144) (0.0474)
1993 1.2418 1.0038
 (0.2059) (0.0492)
1994 0.1117 1.0396
 (0.2974) (0.0522)
1995 1.2674 0.9879
 (0.1697) (0.0520)
1996 0.8273 1.0129
 (0.2178) (0.0525)
1997 0.8840 1.0257
 (0.1912) (0.0544)
1998 1.1198 1.0285
 (0.1457) (0.0550)
1999 0.4886 1.0629
 (0.2776) (0.0520)
2000 0.8887 1.0652
 (0.2519) (0.0555)
2001 1.2008 0.9645
 (0.1599) (0.0527)
2002 1.1278 0.9388
 (0.1612) (0.0502)
2003 1.1011 0.9511
 (0.1411) (0.0494)
2004 1.1579 1.0118
 (0.0958) (0.0470)
2005 0.9594 1.0476
 (0.1427) (0.0481)
2006 1.1553 1.0720
 (0.1235) (0.0478)
2007 0.7027 1.1380
 (0.1846) (0.0482)
2008 1.1606 1.0100
 (0.1313) (0.0474)
2009 0.9954
 (0.0491)

Source: Authors' calculations using SOI data.

(a.) Estimates of equations A.1 through A.4 using a minimum distance
estimator (see section V.Q. Boot-strap standard errors based on 200
replications are in parentheses.

(b.) See appendix D for specification of the polynomial.

(c.) Panel reports estimates of parameters 0 (for the persistent
component) and n (for the transitory component) corresponding to each
year of the sample period (1987-09); parameters are normalized to
equal 1 in 1987 (see appendix D).

APPENDIX B

KSS and GM Methods

Let [[xi].sup.i.sub.t] be residual log income, where t is the calendar year, and where the age index a is suppressed for convenience. In the KSS methodology, the persistent variance in year t is var ([1/P] [[summation].sup.t+k.sub.j=t-k][[xi].sup.i.sub.j]), where k= (P - 1)/2, and where the variance is computed across all individuals (or households) for whom [1/P] [[summation].sup.t+k.sub.j=t-k][[xi].sup.i.sub.j] is defined for a given t. The transitory variance at t is var([[xi].sub.it] - [1/P] [[summation].sup.t+k.sub.j=t-k][[xi].sup.i.sub.j]). Following Kopczuk, Saez, and Song (2010), we set P = 5.

In the GM methodology, let N be the number of individuals, [T.sub.i] [less than or equal to] P the number of years (within the P-year window) that person i is observed, [bar.[[xi].sub.i]] the person-specific average residual log income over [T.sub.i] years, [??] the mean of residual log income across the full sample, and [bar.T] the mean years covered by the window over the individuals in the sample. Then, the exact formula (within each fixed-size window) for the transitory variance is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and for the persistent variance is [1/N - 1][summation].sup.N.sub.i-1][([bar.[[xi].sub.i]] - [xi]).sup.2] - [[[??].sup.2.sub.[upsilon]]/T].

The persistent and transitory variances from GM are similar, although not identical, to the KSS ones. The main difference lies in the presence of the term--([[??].sup.2.sub.[upsilon]]/T) in the persistent GM variance (see Gottschalk and Moffitt 2009, footnote 2).

Note that Gottschalk and Moffitt use P = 9 (rather than our P = 5 in the main text). This slightly reduces the share of the total variance attributed to the persistent component, and slightly increases the share attributed to the transitory component, but has no effect on the trends of the two components.

APPENDIX C

Estimation of the Error Components Model

This appendix provides details of our minimum distance estimator. As mentioned in the text, the estimator matches the model's theoretical variances and autocovariances (specified in levels) to their empirical counterparts. In particular, given any triplet (a, t, k) of normalized age a, calendar year t, and lead k, the error components model in equations 6 through 9 implies a specific parametric form for each autocovariance of residual income, such as cov([[xi].sub.a,t], [[xi].sub.a+k,t+k]). For instance, for (a = 2, t = 1995, k = 0), this would be the variance (since k = 0) in the incomes across all individuals of age 26 in year 1995. These theoretical variances and autocovariances, denoted by cov(a, t, k), are functions of the model parameters [[sigma].sup.2.sub.[alpha]], [psi], [[sigma].sup.2.sub.[eta]], [[sigma].sup.2.sub.[epsilon]], [[theta].sub.1], and [[theta].sub.2], and [[lambda].sub.t], and [[pi].sub.t] for t = 1987, ..., 2009. We estimate these model parameters by minimizing the distance between, on the one hand, the theoretical variances and autocovariances implied by the model, and on the other, their empirical counterparts, which we compute from our longitudinal tax return data for a = 1, ..., 36; t = 198,7 ..., 2009; and k = 0, ..., 22. This yields 7,912 variances and autocovariances that are matched in estimation. Our minimum distance estimator uses a diagonal matrix as the weighting matrix, with weights equal to the inverse of the number of observations used to compute each empirical statistical moment. (49) We do not use an optimal weighting matrix, for reasons discussed in Altonji and Segal (1996).

APPENDIX D

Moment Conditions

Let a be "normalized age" or "potential experience," defined as a = age - 25 + 1, or years starting with age 25. Then, the theoretical moments implied by our baseline error components model in equations 6 through 9 are as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where 1[] is an indicator function equal to either zero or 1.

For t = 1987, 2 [less than or equal to] a [less than or equal to] 36,

var([p.sub.a,1987]) = [[sigma].sup.2.sub.[eta]] [1 - [[psi].sup.2a]/1 - [[psi].sup.2]].

For 1987 [less than or equal to] t [less than or equal to] 2009, a = 1,

var([p.sub.1,t]) = [[sigma].sup.2.sub.[eta]].

For 1988 [less than or equal to] t [less than or equal to] 2009, 2 [less than or equal to] a [less than or equal to] 36,

var([p.sub.a,t]) = [[psi].sup.2] var([p.sub.a-1,t-1]) + [[sigma].sup.2.sub.[eta]].

To obtain identification, we impose the normalization [[lambda].sub.t] = [[pi].sub.t] = 1 for all calendar years t [less than or equal to] 1987, where 1987 is the first year in the sample. Parameter [[lambda].sub.t] (normalized) is restricted to lie on a fourth-order polynomial of the following form: for 1988 [less than or equal to] t [less than or equal to] 2009, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where [??] = t - 1987. (50)

APPENDIX E

Sample Age Distribution by Calendar Year

Sample Age Distribution by Calendar Year

 Age (years) (a)

 Male earnings All households
 sample sample (b)

Year Mean SD Mean SD

1987 39 9.9 39 10.0
1988 39 9.8 39 10.0
1989 39 9.8 39 9.9
1990 39 9.7 40 9.8
1991 39 9.6 40 9.8
1992 40 9.7 40 9.8
1993 40 9.6 40 9.7
1994 40 9.6 40 9.7
1995 40 9.6 40 9.8
1996 40 9.6 40 9.8
1997 40 9.6 41 9.8
1998 41 9.7 41 9.8

 Age (years)

 Male earnings All households
 sample sample (b)

Year Mean SD Mean SD

1999 41 9.6 41 9.8
2000 41 9.6 41 9.8
2001 41 9.7 41 9.9
2002 41 9.7 41 9.9
2003 41 9.7 42 10.0
2004 41 9.8 42 10.0
2005 41 9.9 42 10.1
2006 41 10.0 42 10.2
2007 41 10.0 42 10.2
2008 42 10.1 42 10.3
2009 42 10.1 42 10.3

Source: Authors' calculations using SOI data.

(a.) SD = standard deviation.

(b.) Age is that of the primary filer.

ACKNOWLEDGMENTS We are grateful to the editors, to our discussants Greg Kaplan, Lindsay Owens, and David Grusky, and to Chris Carroll for extremely useful feedback and suggestions. We thank Joe Altonji, Eric Engen, Michael Golosov, Michael Palumbo, Emmanuel Saez, Dan Sichel, and Paul Smith for very helpful comments and discussions. We also thank the participants at the Brookings Panel and at numerous other seminars and conferences. The views presented here are solely those of the authors and do not necessarily represent those of the Treasury Department, the Board of Governors of the Federal Reserve System, or members of their staffs. The authors report no relevant conflicts of interest.

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Comments and Discussion

COMMENT BY GREG KAPLAN

This paper by Jason DeBacker and coauthors provides a new perspective on the much-documented rise in income inequality in the United States, by exploiting confidential data on labor earnings and household income from the Internal Revenue Service (IRS). The IRS data contain information from a large panel of tax returns over the period from 1987 to 2009. The authors use these data to ask whether the recent rise in inequality is mostly due to persistent or to transitory factors. Other authors have answered this question using survey data, predominantly from the Panel Study of Income Dynamics (PSID), and for earlier periods. But this paper breaks new ground in its use of high-quality administrative data to decompose the rise in inequality in the 1990s and 2000s.

DeBacker and his coauthors reach a stark conclusion: all of the recent rise in inequality in male earnings is due to persistent factors; transitory factors have made no contribution to the increase in inequality. Their findings for total household income are similar but less extreme. The authors reach these conclusions using two different approaches. First, they employ simple nonparametric methods, which effectively measure the persistent component of income as a rolling average of income in a given number of adjacent years, and the transitory component as the residual from this rolling average. Second, they estimate error components models (ECMs) for earnings. The ECM approach involves specifying and estimating the parameters of a time-varying stochastic process for income. The persistent and transitory components are then inferred from the estimated model. The authors' conclusions about the relative importance of persistent versus transitory factors are consistent across the two methods.

In this discussion I will elaborate on three issues that are related to these findings, focusing exclusively on the ECM analysis of male labor earnings. First, I will use data from the PSID to investigate how the particular choice of ECM framework may have influenced the authors' conclusions. In doing so I will distinguish between factors that are fixed at the time of entry into the labor market, and shocks that are realized after entry. I will attempt to shed light on which of these factors is responsible for the increase in the persistent variance. I will also explain how an increase in the variance of shocks that occurred before 1987 could be responsible for the observed increase in inequality from 1987 to 2009 even in the absence of any changes in the labor market during this period. Second, I will use the PSID data to investigate the importance of changes in the returns to education in accounting for the authors' findings. I will show that the findings are mostly consistent across the two data sets and are not substantially affected by controlling for education. Third, I will highlight an issue that the authors do not address, but that is a natural one to raise in light of their findings, and given their access to the IRS data: in which part of the income distribution is the recent rise in inequality concentrated? I will conduct a simple decomposition using the PSID data to investigate this issue.

How do the publicly available PSID data compare with the confidential IRS data used by the authors? The baseline sample of male earners from the IRS contains 221,099 person-year observations on 20,859 individuals over the period 1987-2009. In all of the analyses that follow, I use a sample of male heads of households from the PSID that imposes the same selection criteria for age and minimum annual earnings as the authors impose on the IRS data. The resulting sample contains 70,479 person-year observations on 6,778 individuals over the period 1970-2008 (the data are biennial after 1996). Thus, the IRS sample is about three times the size of the PSID sample, both in terms of individuals and in terms of individual-year observations.

My figure 1 plots inequality in male earnings, as measured by the standard deviation of the logarithm, in the two data sets over time. For the period over which the two samples overlap, the trends in inequality are very similar. The level of inequality is about 0.1 log point higher in the IRS data, likely because of undersampling of very high earners in the PSID. Moreover, the IRS series appears far less noisy than the PSID series, which reinforces the view that the IRS data are useful for reevaluating questions that have been addressed using PSID data in the existing literature, such as the cyclicality of idiosyncratic labor income risk (compare, for example, the difference in the increase in inequality during the 1990-91 recession in the two series in this figure).

Figure 1 also puts in perspective the magnitude of the rise in inequality that DeBacker and coauthors decompose. Although inequality has undoubtedly increased between 1987 and 2009 in the IRS data, the magnitude of the increase is smaller (about 0.05 log point) than that in the 1970s and 1980s in the PSID data (about 0.15 log point). Both data sets have advantages and disadvantages. The IRS data set is cleaner and larger and has better coverage at the top of the earnings distribution. Yet it is confidential and lacks data on demographic information, such as education. The PSID data, on the other hand, are publicly available and contain many demographic and financial variables.

[FIGURE 1 OMITTED]

The ECM framework that DeBacker and his coauthors employ is one of many possible choices. Consider the following parametric model for residual log earnings of individual i in year t, [[xi].sup.i.sub.t]:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[epsilon].sup.i.sub.t] and [[eta].sup.i.sub.t] are mean-zero i.i.d (over time) shocks with constant variances [[sigma].sup.2.sub.[epsilon]] and [[sigma].sup.2.sub.[eta]], and [[alpha].sup.i] is a mean-zero fixed effect with variance [[sigma].sup.2.sub.[alpha]]. The authors refer to the component ([[alpha].sup.i] + [p.sup.i.sub.t]) as the persistent component and to [[tau].sup.i.sub.t] as the transitory component. I will adopt the same terminology. The process in equation 1 differs from the one in the paper only in that the transitory component is modeled as an MA(1) rather than an MA(2) process. This difference is not consequential and helps to simplify the analysis.

Decomposing changes over time in the variance of residual log earnings requires allowing some or all of the parameters in equation 1 to change over time. There are many ways to do this. One natural way, which I will refer to as version A, allows the variances of the two shocks to change over time, and the price of the fixed effect to change over time. Thus, in version A the variances of the two shocks become [[sigma].sup.2.sub.[epsilon]t], and [[sigma].sup.2.sub.[eta]t], and the first line in equation 1 is modified to read

(2) [[xi].sup.i.sub.t] = [[lambda].sub.[alpha],t] [[alpha].sup.i] + [p.supp.i.sub.] + [[tau].sup.i.sub.t]

where a normalization is imposed on [[lambda].sub.[alpha],0]. In this interpretation the ECM changes over time for two reasons: individuals experience persistent and transitory shocks that are drawn from a more or less dispersed distribution, and the market price of an individual's fixed skills is changing over time.

Figure 2 shows the results from estimating ECM version A using the PSID data. The estimate of the autoregressive parameter, [psi], is 0.962, and the estimate of the moving average parameter, [theta], is 0.215. To keep the procedure as close as possible to that in the paper, I have restricted the price of skills, [[lambda].sub.[alpha],t], and the variance of persistent shocks, [[sigma].sup.2.sub.[eta]t], to lie on fourth- degree polynomials in r The variance of the transitory shock, [[sigma].sup.2.sub.[epsilon]t], is left unrestricted. Consistent with a large existing literature, the estimates reveal that the variance of persistent shocks increased from the late 1970s to the late 1980s, but was then constant until the mid-2000s before starting to rise again. The variance of the fixed component, [[lambda].sup.2.sub.[alpha],t][[sigma].sup.2.sub.[alpha] also increased during the 1970s and 1980s, but then declined substantially in the 1990s and early 2000s.

The implied variance of the total persistent component, [[alpha].sup.2.sub.[alpha],t] + var([p.sup.i.sub.t]), is shown in the left-hand panel of figure 3. The PSID estimates of ECM version A suggest that the variance of the persistent component of income increased sharply from 1975 to 1990, but was flat (or declined slightly) between 1990 and 2005. After 2005 the variance began to increase again. The behavior of the variance of the persistent component in the 1990s contrasts with DeBacker and coauthors' finding of an increase in the 1990s in the IRS data. Yet given the estimated variances in figure 2, one might be surprised that the PSID estimates do not reveal an even larger decline in the variance of the persistent component: that graph shows that the 1990s were a period with no increase in the variance of persistent shocks, while the variance of the fixed component declined substantially. The reason why the variance of the total persistent component does not decline more is that even though there was no increase in the variance of persistent shocks during this period, the earlier increases in [[sigma].sub.[eta]], during the 1980s led to a continued increase in the variance of the persistence component, var([p.sup.i.sub.t]), well into the 1990s. This occurs because it takes time for the older cohorts who were subject to the small shocks of the 1970s to be replaced by the younger cohorts who were subject to large shocks for their entire working life.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

The cohort effect that arises from changes in [[sigma].sub.[eta]t], is something to bear in mind when interpreting the findings in this paper. The IRS data begin only in 1987, which is exactly when the variance of the persistent shocks levels off in the PSID data. Thus if, as one might expect, there was also an increase in [[sigma].sub.[eta]t] before 1987 in the IRS data, one would expect to see an increase in the variance of the persistent component in the 1990s. This increase would not be due to changes that occurred after 1987, yet estimation using the authors' strategy with IRS data would necessarily attribute the increase to a change that occurred after 1987, since their framework cannot handle lagged effects of pre-1987 changes. Unfortunately, little can be done about this given the available data, and a similar criticism might apply to the PSID estimates regarding changes that occurred before 1970.

An alternative way to allow the parameters in equation 1 to change over time is to fix [[sigma].sub.[eta]] but modify the first line of the equation to read

(3) [[xi].sup.i.sub.t] = [[lambda].sub.[alpha],t][[alpha].sup.i] + [[lambda].sub.p,t][p.sup.i.sub.t] + [[tau].sup.i.sub.t].

I will refer to this model as ECM version B. Here the interpretation is that the dispersion of the persistent shocks that hit individuals does not change over time. Instead the accumulation of these shocks, [p.sup.i.sub.t], is interpreted as slow movement in a stock of individual-specific human capital or skills, which command a price in the labor market [[lambda].sub.p,t]. The price of these skills is allowed to change over time, which leads to changes in the cross-sectional variance of residual earnings. The conceptual distinction between [[alpha].sup.i] and [p.sup.i.sub.t] in this interpretation is that [[alpha].sup.i] reflects skills that are determined at the time of entry into the labor market, whereas [p.sup.i.sub.t] reflects skills that continue to evolve stochastically after entry. Finally, one could also impose the restriction that [[lambda].sub.[alpha],t] = [[lambda].sub.p,t] = [[lambda].sub.t], so that the first line of equation 1 reads

(4) [[xi].sup.i.sub.t] = [lambda]([[alpha].sup.i] + [p.sup.i.sub.t]) + [[tau].sup.i.sub.t].

I will refer to this model as ECM version C. Here the interpretation is that the market does not distinguish between the value of skills obtained before entry into the labor market (such as formal education) and the value of skills acquired later in life (such as on-the-job training or job-specific human capital). This is the interpretation that the authors adopt, since version C is the specification that the authors estimate with the IRS data.

How does the choice of ECM affect one's conclusions about the rise in the persistent variance of earnings? My figure 3 attempts to answer this question by reporting estimates of versions B and C from the PSID as well as of version A. The left-hand panel shows that the variance of the total persistent component is essentially identical in all three versions (the three versions also deliver very similar estimates for the autoregressive and the moving-average parameters). Thus, to the extent that these findings carry over to the IRS data, it is unlikely that the authors' conclusions about the rise in the variance of the total persistent component would have been changed by adopting either version A or version B.

Although the three versions of the ECM yield the same estimates over time for the variance of the total persistent component, they yield very different estimates for how this variance is divided between factors that are fixed at the time of entry to the labor market, [[alpha].sup.i], and factors that evolve stochastically over time, [p.sup.i.sub.t]. These differences are illustrated in the right-hand panel of figure 3, which shows the variance of the fixed effect, [[lambda].sup.2.sub.[alpha],t][[sigma].sup.2.sub.[alpha],t], for each of the three versions. Version A, which allows for the size of persistent shocks to change over time, attributes a much bigger role to movements in the price of fixed skills in accounting for changes in the variance of the persistent component, compared with either version B or version C. The distinction between cross-sectional variation in earnings due to fixed factors and variation due to the realization of shocks is potentially important. First, the two views of the increase in earnings inequality may have different implications for the increase in consumption inequality (and thus welfare) in a structural life cycle model of intertemporal consumption choice, since the impact of the changes in [[lambda].sub.t], depends crucially on the assumptions one makes about how these changes enter workers' information sets. Second, the appropriate policy interventions for influencing the earnings distribution are different: the latter view points to the importance of labor market interventions, whereas the former points to education interventions.

[FIGURE 4 OMITTED]

Cross-sectional variation in the fixed effect, [[alpha].sup.i], is partly due to cross-sectional differences in observed education and partly due to cross-sectional differences in unobserved cognitive and noncognitive skills. Given the importance of changes in [[lambda].sub.t] in accounting for the change in earnings inequality in the IRS data, it is natural to ask whether these changes reflect an increase in returns to traditional measures of education or an increase in returns to the unobserved components of skills. This question cannot be answered with the IRS data, but it can be answered with the PSID data. To address this, my figure 4 presents estimates using data on residual log earnings, [[xi].sup.i.sub.t], that are constructed in two different ways. The lines labeled "without education controls" are estimates based on data where [[xi].sup.i.sub.t] is constructed as the residual from a regression of log earnings on a full set of age dummies in each year. This is the same approach followed by DeBacker and coauthors. The lines in figure 4 labeled "with education controls" are estimates based on data where [[xi].sup.i.sub.t] is constructed as the residual from a regression of log earnings on a full set of age dummies, education dummies, and education x age interactions in each year.

The left-hand panel of figure 4 displays parameter estimates of ECM version A with and without education controls. Both the estimates of the variance of the fixed effects [[lambda].sup.2.sub.[alpha],t][[sigma].sup.2.sub.[alpha]] and the variance of persistent shocks [[sigma].sup.2.sub.[eta]t] are affected by the education controls. At least half of the increase in the variance of the fixed effects and the subsequent decline between 1970 and 2000 is due to returns to education, but the increase in the variance in the 2000s is the same in both specifications. This result is useful in interpreting DeBacker and coauthors' findings, since they cannot control for education in the IRS data. Using the PSID findings as a guide, one might conclude that the recent increase in the market price of skills that the authors document would remain largely unchanged if they were able to control for education. It appears that the increase is driven by an increase in the returns to unobserved skills rather than returns to formal education.

The right-hand panel of figure 4 offers an alternative perspective on the likely effect of controlling for education on DeBacker and coauthors' findings, by estimating ECM version C (the authors' preferred specification) with and without education controls on the PSID data. These estimates also indicate that the biggest differences in trends under the two specifications occur before the 1990s, further reinforcing the view that the increase in the variance of the persistent component in the IRS data reflects an increase in returns to unobserved skills within education groups.

Before concluding, I will raise one additional issue that the authors do not tackle, but that could be addressed with their IRS data. The authors focus their analysis on determining whether the recent increase in earnings inequality has been persistent or transitory in nature, and conclude that it is entirely the former. In addition, one might ask which individuals have been most affected by this increase in the variance of the persistent component. Specifically, many researchers and policymakers are interested in understanding whether changes in inequality affect mostly high-earnings individuals, low-earnings individuals, or individuals in the middle of the income distribution. The IRS data set is well suited to address this issue, again because it is larger and cleaner than the PS1D (particularly at the top of the distribution). One possible approach to answering this question is to decompose the cross-sectional variance of log earnings (or residual log earnings), [y.sup.i.sub.t], in each year as follows:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

where [r.sup.i.sub.t] is the rank of individual i in the year t earnings distribution. The first two terms in equation 5 are the variances of earnings within the bottom and the top half of the earnings distribution, respectively. The third term in equation 5 is the component of the variance of earnings that is due to the difference in average earnings between the top half and the bottom half. The decomposition here focuses on the overall cross-sectional variance, but the panel nature of the IRS data lends itself to a similar decomposition of only the persistent component of earnings, for example by first employing the authors' simple nonparametric methods.

[FIGURE 5 OMITTED]

My figure 5 displays the results from implementing this decomposition in the PSID. All three components are normalized to 1 in 1970. The figure shows that since the early 1980s, there has been essentially no increase in the variance of earnings in the bottom half of the distribution. By contrast, the variance within the top half of the distribution has increased steadily since 1980 and continues to rise. The gap between average earnings in the two halves of the distribution has also continued to widen in recent years. Thus, the PSID data suggest that there are important asymmetries in the earnings distribution and that the recent increase in inequality is a more complicated phenomenon than just changes in dynamics of the first and second moments of the earnings process.

Given these asymmetries, a useful step forward for the literature would be to move toward richer, possibly nonlinear, models of earnings dynamics that can shed light on the complicated changes in the earnings distribution observed in recent years. This paper is a useful starting point. The IRS data set, a large panel of earnings data that is mostly free of measurement error and top-coding, is an ideal resource for such an investigation. Efforts to further improve this data set could lead to large benefits for researchers, policymakers, and ultimately the welfare of individuals. Such efforts might be focused on extending the sample back before 1987 or on making a suitably anonymized version of the data available for wider use.

COMMENT BY LINDSAY A. OWENS and DAVID B. GRUSKY (1)

It has long been argued that the ongoing increase in income and earnings inequality cannot be well understood until it is decomposed into persistent and transitory components. The persistent component pertains to the inequality generated by the permanent characteristics of individuals (their education, unobserved ability, and the like), whereas the transitory component pertains to the inequality generated by temporary shocks (such as a temporary illness, transitory unemployment, or a change in jobs). It is not implausible that the takeoff in income inequality partly reflects the emergence of a labor market that is increasingly subject to transitory shocks in the form of a growing risk of unemployment, underemployment, or job change. If this is indeed the case, it might change our understanding of both the sources of the takeoff and its implications for social welfare.

The key contribution of Jason DeBacker and his coauthors in this paper is to bring a large panel of tax returns to bear on this debate. The results reveal that the entire rise in inequality in male earnings, and most of that in household income, is attributable to an increase in the dispersion of the persistent component.

We leave it to others to comment on the models, the data, and other technical features of this analysis. It suffices for our purposes to stress that the analysis is noteworthy because of the extraordinary data upon which it rests. The confidential panel of Internal Revenue Service (IRS) tax returns delivers unusually high-quality earnings and income data for an unusually large sample. Moreover, the authors apply an impressive range of parametric and nonparametric approaches to the IRS data, with reassuringly similar results. The authors also supplement the more conventional and usual analyses of earnings data with revealing analyses of pre-tax and after-tax household income. For all of these reasons, the authors have contributed an important paper, and their results merit close attention.

We are so impressed with the paper that we are inclined to stipulate that it is a major contribution, forgo the usual internal critique, and instead take on the task of considering how the analyses might be usefully elaborated upon in light of the opportunities that the IRS data open up. We approach this question from the point of view of better understanding the welfare implications of inequality. The long-standing presumption in this regard is that, insofar as the takeoff in inequality is mainly generated by an increase in transitory shocks, it is less consequential for welfare because individuals can always borrow against future income and smooth out the effects of such shocks. The takeoff in inequality might therefore be understood from a welfarist stance as entailing little more than the nuisance of engaging in more smoothing than had before been necessary.

This comment will consider whether considerations of welfare are indeed adequately understood in these terms. We first suggest that a welfarist stance, if rigorously adopted, instead leads us to privilege the concept of lifetime income and to move toward IRS-based analyses of trends in lifetime income. We next argue for extending the characteristic focus on intraindividual transfers to a more encompassing consideration of interindividual transfers.

THE CASE FOR A LIFETIME INCOME APPROACH The simple point with which we begin is that, insofar as one is willing to assume away liquidity constraints that prevent smoothing, it seems appropriate to do so wholeheartedly and move directly to analyzing data on lifetime income. The obvious virtue of this approach is that it obviates the need to parameterize the potentially complicated ways in which a shock may or may not have short-term or long-term effects. If, for example, a lottery winner decides to immediately exit the labor market, this decision will ultimately be revealed in his or her lifetime income. The same applies to such shocks as pregnancy, unemployment, job shifting, or receipt of program benefits (such as the earned income tax credit, food stamps, or unemployment insurance). Although the authors very elegantly model how the income effects of such shocks tend to dissipate over time, an attractively nonparametric alternative is simply to examine trends in the inequality of lifetime income, an approach that is approximately equivalent to applying the method of Kopczuk, Saez, and Song (2010) with a very large P parameter (where P refers to the number of years over which income is averaged).

What makes this nonparametric approach attractive? If one cares about the welfare implications of inequality, surely the first cut at understanding those implications is to examine the first moment of each individual's own distribution of income across years. The presumption, in other words, is that individuals operating under a veil of ignorance about their own distribution of future annual earnings would, more than anything else, want to know how much they will make on average per year (as well as the number of years they will have earnings). It follows that the inequality of those lifetime averages, calculated separately for each birth cohort, would speak rather directly to matters of welfare, arguably more directly than any of the parametric or nonparametric approaches deployed in this paper. That said, we well appreciate that conventional parametric and nonparametric approaches are useful for a host of other objectives, including making inferences about consumption and consumption inequality. It must also be conceded that a lifetime income approach implies a rather delayed reading of trends, because each birth cohort enters the series only after its members complete their labor force participation. This is clearly a disadvantage insofar as real-time reporting is desired. We are merely suggesting that a lifetime income approach is but one additional tool that happens to be especially useful when one is making judgments about welfare.

It bears noting that such an approach entails a shift of emphasis from period analyses to cohort analyses of trends in income inequality. This shift is attractive because it allows one to better capture the effects of forces that operate in cohort-specific ways. For example, recessions have especially prominent effects on birth cohorts that come of age during the recession itself, and these effects in turn serve to suppress lifetime earnings (see Kahn 2010). Insofar as recessions are inequality enhancing (because they hit poorly credentialed workers the hardest), a cohort approach will reveal that effect especially clearly. There is good reason to believe that other important sources of the trend in inequality (such as changes in schooling institutions or early-childhood antipoverty interventions) likewise operate in cohort-targeted ways that will be obscured by the field's typical emphasis on period effects.

It is also attractive to focus on cohorts because the invidious comparisons that individuals make tend to feature their same-age peers. As life unfolds, individuals compete in schools and in the workplace with members of their own birth cohort, and the outcome of that age-specific competition is likely to affect self-assessments. We expect, for example, that individuals will be more troubled and jealous when they see their same-age peers benefiting disproportionately from the takeoff than when members of some distant birth cohort are the principal beneficiaries. It follows that a cohort approach is especially relevant to considerations of welfare insofar as social comparison processes and their subjective fallout are taken into account.

THE INEQUALITY-EXAGGERATING EFFECTS OF INTERINDIVIDUAL TRANSFERS For those interested in making judgments about welfare, lifetime income is of interest because it is assumed that, without any constraints on liquidity, individuals can freely borrow against their future income stream or freely draw on savings from past streams. This form of borrowing or saving may be understood as an intraindividual transfer from the past or future to the present. If the transitory variance has risen substantially, as some (Heathcote, Perri, and Violante 2010) have claimed, then the takeoff is presumably less troubling because such transfers can smooth out these transitory shocks. The literature has thus focused on the possibility that the usual cross-sectional analyses may overstate the welfare consequences of rising inequality.

The purpose of this section is to shift the focus to various types of interindividual transfers that, if properly taken into account, may lead to the conclusion that the welfare consequences of the takeoff are in fact worse than is usually supposed. That is, whereas a consideration of intraindividual transfers may lead one to overstate the welfare costs of the takeoff in inequality, a consideration of interindividual transfers leads to precisely the opposite conclusion. We develop this argument by considering the welfare effects of interindividual transfers

between spouses, among households within a neighborhood, and between parents and children.

Interspousal transfers To illustrate the argument, we begin by considering the well-known tendency of spouses to pool income, a type of interindividual transfer that motivates the field's long-standing interest in analyzing household or family income inequality. This pooling will increase inequality insofar as there is some amount of income-based "marital homogamy" in which high-income men tend to marry high-income women. In the United States, this form of homogamy is intensifying over time (Schwartz 2010, Mare and Schwartz 2006), a development that contributes to the takeoff in inequality. As Schwartz (2010) reports, the correlation between the earnings of spouses almost tripled between 1967 and 2003, leading in turn to an approximately 25 percent rise in the earnings inequality of families (Schwartz 2010). Although conventional analyses of individual income inequality will not reflect this transfer-based source of rising inequality, there is, of course, a long tradition of analyzing family or household inequality (in which the effects of such homogamy are "built in").

It is striking, however, that critics of conventional cross-sectional analyses of individual income inequality often complain about the possible inequality-exaggerating effects of ignoring intraindividual transfers without acknowledging the opposing inequality-suppressing effects of ignoring interindividual transfers (within households). If one type of transfer-induced bias is to be corrected, then surely the other, opposing bias should be corrected as well. This selective acknowledgment of "transfer bias" cannot be explained by differences in the reliability with which such transfers can be effected. To the contrary, spouses tend to pool income relatively freely on the basis of informal agreements (see Bennett 2013), whereas individuals typically have to engage more formally with friends, parents, or financial intermediaries when seeking to borrow from these sources against their future income. The resulting constraints on liquidity can be substantial (Blank and Barr 2009). This suggests that, if anything, the bias arising from ignoring interindividual transfers should be more troubling than that arising from ignoring intraindividual transfers.

Interneighbor transfers The example of transfers between spouses is, of course, well known. What is perhaps less appreciated is that residential neighbors also engage in pooling and that, by virtue of rising residential segregation, this pooling is leading to a more unequal distribution of valued goods (Reardon and Bischoff 2011). The key dynamic here is again a growth in segregation. That is, just as spouses have increasingly similar incomes (marital homogamy is rising), so too neighborhoods are becoming increasingly homogeneous by income (residential segregation is rising). This means that high-income families are increasingly likely to be living in high-income neighborhoods that give them indirect access to the considerable resources of their neighbors. Because neighborhood goods are often financed by property taxes, it is advantageous to live with high-income neighbors who will contribute substantially to schools, parks, police protection, fire protection, local government, and other public goods. The ongoing takeoff in residential segregation means that this particular advantage, like the advantage of marrying a high-income spouse, increasingly accrues to those with relatively high incomes themselves. This advantage is concealed in conventional analyses of individual income inequality because the "income" takes the form of in-kind resources.

The analogy between these two types of interindividual transfers is by no means perfect. Most obviously, in the United States one makes no overt payment (no dowry) for the privilege of marrying a high-income spouse, whereas one does overtly pay for the privilege of living in a high-income neighborhood. It is accordingly possible that, as neighborhoods become increasingly income segregated, the resulting interneighborhood differences in public goods advantages come to be reflected in the purchase price of homes, thus complicating any effort to understand the effects of this rising segregation on inequality. The second main difference is that spouses typically engage in quite substantial income pooling, whereas residential neighbors are far less collectivist, in effect pooling their income only for a relatively small number of local public goods. The total effects of interneighbor transfers on inequality are, as a result, likely to be comparatively limited.

Intergenerational transfers The third type of interindividual transfer of interest occurs between generations of a family as well as between relatives of the same generation (such as siblings). This type of transfer is closely related to the previous two: it may be understood either as entailing transfers among members of a "virtual neighborhood" defined by kinship ties, or as entailing transfers among members of a "virtual household" that extends beyond those actually living together. Under either interpretation, the key force at work is again rising segregation, which now expresses itself as growing intergenerational elasticities of income. This force, if indeed it is at work, implies that the offspring of high-income families are increasingly likely to find themselves ensconced in virtual households that provide them with access to high-income parents, high-income grandparents, and high-income siblings. It is unclear, however, whether such elasticities are indeed increasing. In a recent review, Chul-In Lee and Gary Solon (2009) conclude that available estimates on trends in intergenerational elasticities are "highly imprecise" (p. 766), mainly because the available data sets (principally the Panel Study of Income Dynamics) are extremely small.

There is nonetheless good reason to worry that these elasticities are on the rise (see Krueger 2012). If indeed they are, what does it mean for our understanding of trends in income inequality? It suggests that high-income offspring may be more likely to receive gifts or substantial inheritances that then generate investment income. Because these income transfers are at least partly revealed as individual income (among the offspring), they will not be concealed in conventional individual analyses of income inequality. However, many of the transfers again take an in-kind form, such as unreported gifts, access to lavish parental vacation homes, or "parental buffering" of children when they experience unemployment or other labor market difficulties. The provision of such goods will tend to increase inequality insofar as they are disproportionately available to high-income offspring.

CONCLUSION It is testimony to our high regard for the analysis in this paper that, rather than carry out the usual critique of its methods or conclusions, we have instead sought to consider various extensions of their analysis. We began by suggesting that the welfare implications of inequality might be better understood by supplementing the usual parametric approach with a nonparametric analysis of lifetime income inequality. The IRS tax data are well suited to the cohort analysis that such an approach implies.

We have also argued that an exclusive focus on intraindividual transfers may have distracted scholars from appreciating how various interindividual transfers may create inequalities that conventional individual analyses miss. Because high-income individuals are increasingly embedded in networks that provide access to income or in-kind benefits provided by others (spouses, parents, extended families, neighbors), existing models of individual income inequality may understate the welfare implications of rising inequality, a bias that is precisely the opposite of that emphasized by those who attend exclusively to intraindividual transfers. It is unclear why the field has been so captivated by intraindividual transfers when the countervailing effects of interindividual transfers may be more important. The IRS data provide an opportunity to develop models that can at once capture changes in inequality as well as these possible changes in income dependencies within households, neighborhoods, and extended families.

REFERENCES FOR THE OWENS AND GRUSKY COMMENT

Bennett, Fran. 2013. "Researching Within-Household Distribution: Overview, Developments, Debates, and Methodological Challenges." Journal of Marriage and Family 75, no. 3: 582-97.

Blank, Rebecca M., and Michael S. Barr, eds. 2009. Insufficient Funds: Savings, Assets, Credit and Banking among Low-Income Households. New York: Russell Sage.

Kahn, Lisa B. "The Long-Term Labor Market Consequences of Graduating from College in a Bad Economy." Labor Economics 17, no. 2: 303-16.

Krueger, Alan B. 2012. "The Rise and Consequences of Inequality in the United States." Washington: Executive Office of the President. www.whitehouse.gov/ sites/default/files/krueger_cap_speech_final_remarks.pdf.

Lee, Chul-In, and Gary Solon. 2009. "Trends in Intergenerational Income Mobility." Review of Economics and Statistics 91: 766-72.

Mare, Robert D., and Christine R. Schwartz. 2006. "Educational Assortative Mating and the Family Background of the Next Generation: A Formal Analysis." Sociological Theory and Methods 2 l: 253-77.

Reardon, Sean, and Kendra Bischoff. 2011. "Growth in Residential Segregation of Families by Income, 1970 to 2009." US2010 Research Brief. New York: Russell Sage.

Schwartz, Christine R. 2010. "Earnings Inequality and the Changing Association between Spouses' Earnings." American Journal of Sociology 115, no. 5: 1524-57.

(1.) The Stanford Center on Poverty and Inequality is supported by grant number AE00101 from the U.S. Department of Health and Human Services, Office of the Assistant Secretary for Planning and Evaluation (awarded by the Substance Abuse Mental Health Service Administration). The contents of this comment are solely the responsibility of the authors and do not necessarily represent the official views of the U.S. Department of Health and Human Services, Office of the Assistant Secretary for Planning and Evaluation.

GENERAL DISCUSSION John Haltiwanger noted that job destruction rates and unemployment inflow rates had declined over the authors' study period. If job flows and unemployment are treated in the model as transitory shocks to income, those trends should be driving the temporary component in income inequality downward, to the point where the permanent component alone might account for, or more than account for, the observed results.

William Brainard agreed with the discussants' suggestion that the authors address the differences between their tax data and other data sets in widespread use. He also pointed out that there is substantial heterogeneity in individuals' lifetime income profiles. Some occupations have a period of apprenticeship, which causes the profile for those workers to be flat initially; unionized workers, in contrast, have a very different pattern. Because the authors' model does not account for these individual differences, Brainard thought, all of them would show up in the permanent component, when in fact they are caused by interaction with the individual's education and other factors. Brainard suggested that the authors take the structural differences between individuals more thoroughly into account by including age and education covariates.

Justin Wolfers requested that the authors clarify how they distinguished between permanent and transitory shocks. In reply, Greg Kaplan described the method with reference to a random walk model. In each period a shock occurs that either increases or decreases the individual's income. The sum of these shocks over time was taken to be the permanent component for the individual, and a time-varying, universal weighting factor was applied to that sum. The more traditional method, Kaplan noted, would be to apply the weighting factor to the individual shocks rather than to their sum.

Christopher Carroll called the authors' method an interesting innovation but observed that, since their data were also novel, it was difficult to determine what portion of the difference between their results and those of others working in this area was being driven by their modeling choice and what portion by their novel data set. He urged the authors to go back and apply the simplest standard model to their data, to serve as a benchmark, and from there do further analysis to see what cannot be explained by that simple model.

Carol Graham noticed an upward tick in permanent income inequality and a downward one in transitory inequality in the authors' data around 2007. She wondered whether those movements represented merely transient phenomena or whether what was happening in that period might explain some of the difference between permanent and transitory income.

Replying to a comment made by David Grusky in his formal discussion, Robert Gordon questioned whether the income homogeneity of neighborhoods is in fact increasing. His own impression was that demographic changes have been making income more heterogeneous as blacks move into the suburbs and back to the South while Hispanics and affluent whites move into the inner cities. New York's East Village, for example, was uniformly poor 30 years ago, but more recently the boom in the city's financial services and entertainment industries had brought some very wealthy people into the neighborhood, leading to a mix of incomes. Gordon also challenged Grusky's implication that a decline in intergenerational mobility was not yet in evidence. He cited recent findings that the United States today has one of the lowest levels of intergenerational mobility among developed economies. One can almost predict, Gordon added, that this trend will persist, as it is being reinforced by the behaviors of those at both the top and the bottom: the wealthy are taking pains to ensure that their children learn foreign languages (and economics), while the share of children in the poorest third of the white population living with both parents continues to decline.

Gita Gopinath commented that an increase in permanent income inequality will have implications for consumption inequality, and thus that looking at consumption decisions should make it easier to determine whether a given income shock is transitory or permanent. Gopinath was curious to know whether the paper's results were driven by the fixed effects or the random walk component of the income shock. The answer, she thought, could help determine whether today's income inequality was caused by a widening difference in payoffs between high- and low-ability workers.

Richard Cooper agreed with Brainard that the authors' distinction between permanent and temporary income was highly suspect. He also thought it would be valuable to compare individuals' reported W-2 (wage) income with their income reported on Schedules C and D (business income and capital gains, respectively) of their IRS Form 1040. That information could help determine how much income inequality is due to proprietary income and how much to earnings from labor. Gordon remarked that a paper by Thomas Piketty and Emmanuel Saez had done just such an analysis and found that the increase in inequality came mainly from labor earnings. A caveat to that finding, however, was that stock options--an important contributor today to incomes at the top--are inappropriately reported as labor earnings.

Responding to the discussion, Ivan Vidangos argued that the distinction between permanent and transitory components was necessarily fairly arbitrary. In the real world shocks can be very transitory, very permanent, or anywhere in between, but one has to draw the line somewhere. Their strategy was to select two points near the ends of the continuum and see if the results differed dramatically. They had experimented with many different specifications, including one that indicated that permanent factors were capturing 87 percent of the variance and another that put it at 36 percent, but in all cases the trends showed that the rise in inequality was driven by the permanent component.

JASON DEBACKER

Middle Tennessee State University

BRADLEY HELM

Indiana University

VASIA PANOUSI

Board of Governors of the

Federal Reserve System

SHANTHI RAMNATH

U.S. Department of the Treasury

IVAN VIDANGOS

Board of Governors of the

Federal Reserve System

(1.) In this study our baseline measure of income inequality is the cross-sectional variance (that is, the variance across all individuals or households in our sample at a given time) in the logarithm of annual income. We use the terms "persistent inequality" and "persistent variance" to refer to the variance of the persistent component of income. Therefore, an increase in inequality is called "persistent" if it is driven by an increase in the variance of the persistent component of income. A similar interpretation will apply to "transitory inequality" and "transitory variance."

(2.) The analysis was conducted at and approved by the U.S. Treasury Department to ensure that the strictest confidentiality is preserved.

(3.) Throughout the paper, we refer to error components models as nonstationary if model parameters are allowed to change over calendar time so as to capture changes over time in the distribution of income (including its dispersion).

(4.) For instance, Kopczuk, Saez, and Song (2010) use longitudinal earnings data from SSA records to document that inequality in annual earnings among men has been rising since around 1970. See also the earlier contributions by Bound and Johnson (1992), Katz and Murphy (1992), Murphy and Welch (1992), Juhn, Murphy, and Pierce (1993), Katz and Autor (1999), and more recently, Autor, Katz, and Kearney (2008).

(5.) Baker and Solon (2003) find broadly similar results for Canada using administrative data.

(6.) Heathcote, Peril, and Violante (2010) document patterns in inequality over time in a number of variables at the individual and the household level. Their decomposition of changes in the variance of earnings into transitory and persistent components is not the main focus of their paper. Also, they use hourly wages, rather than annual earnings, and estimate a simpler error components model. Our approach is closer to that of Moffitt and Gottschalk (2011).

(7.) In our online appendix, however, we present some results suggesting that the transitory component might play more of a role in the PSID data than in administrative data. Online appendixes for papers in this volume may be found at the Brookings Papers website, www.brookings.edu/about/projects/bpea, under "Past Editions."

(8.) Blundell, Pistaferri, and Preston (2008) find an increase in the variance of persistent income shocks in the early 1980s, followed by an increase in the variance of transitory shocks in the late 1980s. We cannot directly compare our results with theirs, as our sample periods barely overlap.

(9.) Dynan, Elmendorf, and Sichel (2012) find a continuous increase in the volatility of male earnings in the PSID over the 1967-2004 period. However, their measure of earnings includes income from self-employment and hence is not directly comparable to ours or to that of the studies mentioned above.

(10.) The fraction of U.S. households filing tax returns is generally around 90 to 95 percent (see, for example, Piketty and Saez 2003). Most households who do not file taxes are low-income households. Therefore, our data might miss some changes in income inequality at the bottom of the income distribution. However, we do not view this as a first-order concern, because, as documented by Autor, Katz, and Kearney (2008) and Kopczuk, Saez, and Song (2010), changes in income inequality over our sample period have been concentrated in the upper part of the income distribution.

(11.) On tax returns in which a married couple is filing jointly, the primary filer is the individual listed first on Form 1040. This is usually, although not always, the husband. On tax returns of single fliers, the primary filer is the individual who filed the return.

(12.) The full 1987 stratified random sample actually consisted of two parts: the random sample mentioned in the text and a high-income oversample. We do not use the high-income oversample in our analysis in this paper.

(13.) In addition, it is well known that changes in income at low levels of income can unduly affect estimates of models of the income process. Two commonly used approaches to address this issue are to exclude low-income observations or to left-censor them. Given the issues discussed above, we choose to exclude them.

(14.) This is the same threshold as used by Kopczuk, Saez, and Song (2010). The threshold equals $2,575 in 2004 and is indexed for other years by nominal average wage growth. In the online appendix we check the sensitivity of our results to setting lower and higher minimum thresholds.

(15.) For household income the figures use our "all households" sample. In our "male-headed households" sample, the cross-sectional variance (of the log) increases by 0.22 squared log point for pre-tax and 0.17 squared log point for after-tax household income.

(16.) Furthermore, in the online appendix we examine the robustness of our results to alternative treatments of household size and composition.

(17.) Indeed, for most specifications of an income process, volatility and the variance of transitory income changes tend to move closely together, although in many cases volatility also captures part of the variance of persistent income changes. See Shin and Solon (2011) for a detailed discussion.

(18.) For 1-year changes the estimated coefficient is 0.00037, with a standard error of 0.00050. This coefficient would imply an increase of less than 0.01 in the standard deviation over 23 years. For 2-year changes the coefficient is 0.00046, with a standard error of 0.00058.

(19.) Note that, by taking averages across periods, this method attenuates somewhat the increase in both persistent and transitory inequality, and thereby in total inequality, constructed here as the sum of its persistent and transitory parts.

(20.) That is, to compute the variance of [[alpha].sub.i] and [[epsilon].sub.it] in a given year t, the method treats the data in the P-year window centered around t as if they were the entire data set available.

(21.) The difference between the KSS and GM methods essentially reflects a "bias correction term" in the random effects formula upon which the GM decomposition is based. For the exact formulas used by the GM method, see appendix B. Also see the discussion of the method in Gottschalk and Moffitt (2009).

(22.) The lines in the figure labeled "ECM-predicted" correspond to predicted values from the nonstationary model that we introduce in the next section and are discussed in section V.D.

(23.) More precisely, and as we discuss below, the objective in estimation is to match the entire set of variances and autocovariances that can be computed from the data.

(24.) Stationary, univariate error components models have been estimated in a large number of papers. An incomplete list includes the early contributions of Lillard and Willis (1978), Lillard and Weiss (1979), and MaCurdy (1982). See also Carroll (1992), Baker (1997), Carroll and Samwick (1997), and more recently, Guvenen (2009) and Hryshko (2012). Richer, multivariate stationary models have recently been estimated in Low, Meghir, and Pistaferri (2010) and Altonji, Smith, and Vidangos (forthcoming).

(25.) The index a actually represents "normalized age" or "potential experience," defined as a = age - 25 + 1, or years starting with age 25.

(26.) The covariates [X.sup.i.sub.a,t] used for the g(*) component in these regressions correspond exactly to the discussion in section III. The residuals [[??].sup.i.sub.a,t] obtained from equation 1 are thus identical to the residuals discussed in section III, and equation 1 formalizes their definition. As noted in section III, the regressions are run separately by calendar year.

(27.) For the variance profile, a value of [psi] of exactly 1 would imply an exactly linear increase in the variance of [p.sup.i.sub.a,t] as a function of age. For the autocovariance function, the decline in the covariances after the first couple of years in the model is entirely determined by the value of [psi]. The slow gradual decline seen in the data requires a value of [psi] that is close to, but smaller than, 1.

(28.) See, for example, Meghir and Pistaferri (2004), Baker (1997), MaCurdy (1982), and Abowd and Card (1989).

(29.) Our estimation methodology is discussed in the next section, in the more general context of our nonstationary model, which nests the stationary specifications presented here.

(30.) One difference is that, as should be expected, our estimate of parameter [[sigma].sup.2.sub.[alpha]] is larger than the estimates typically found by studies using residuals that have removed the effects of education.

(31.) As already noted, the lines labeled "ECM-predicted" in figure 4 show the fit of the nonstationary version of this model and are discussed in section V.D.

(32.) Card and Lemieux (1996) provide evidence in support of this idea.

(33.) In appendix A we present results for an alternative nonstationary specification in which the [[lambda].sub.t] parameters multiply the [[alpha].sup.i] component only, and in which the variances of the persistent shocks are allowed to vary over time. The results from that alternative specification are consistent with the results obtained with our baseline model.

(34.) Using a quadratic or a cubic polynomial instead yields similar results. In general, we have found that restricting the [[lambda].sub.t] parameters to lie on a polynomial has little effect on the trend captured by the [[lambda].sub.t] series. The restriction also has little effect on the model's ability to match the trend in the total variance, since the [[pi].sub.t] parameters pick up the transitory part of the variation in the (fully unrestricted) [[lambda].sub.t]. Results for the unrestricted [[lambda].sub.t] are presented in the online appendix and yield similar conclusions.

(35.) The "ECM-predicted" series are constructed in the same way as the "empirical" series, but using the theoretical moments implied by the estimated model rather than the empirical moments.

(36.) We could also use the estimated model to compute similar decompositions for any age group, or for any age distribution. In the online appendix we perform the decomposition assuming a constant age distribution, and the results are essentially unchanged.

(37.) We do not show separately the empirical cross-sectional variance of log residual male earnings because it looks indistinguishable from the top line in figure 5. However, the latter differs somewhat from the variance of log male earnings shown in figure l, because figure 5 uses residuals that have removed the variation in earnings that is due to age, whereas figure 1 uses the raw data.

(38.) Along the same lines, previous versions of this paper included specifications where the transitory component followed an ARMA(1, 1) process, which exhibited more persistence than the transitory component in the ECM model presented above (in those specifications, the [p.sup.i.sub.a,t] component was restricted to a random walk). As suggested by the previous discussion, those specifications attributed a larger share of the total variance to the transitory component at any given point in time, but the results for the trends were essentially identical.

(39.) Using the sample of all households, on average over 1987-2009, male labor earnings account for about 54 percent of total household income, female labor earnings for 26 percent, retirement and transfer income for 5 percent, investment income for 8 percent, and business income for 7 percent.

(40.) It also adds some household observations for which labor earnings of the male filer are below the minimum threshold, but for which total household income is above the minimum threshold.

(41.) In the online appendix we investigate the robustness of our results to alternative treatments of household size and composition.

(42.) Note that the total variance of household income in figure 8 is lower in any given year than the total variance of male earnings shown earlier. The reason is that these are variances of residuals, which in the case of household income have removed all variation explained by household size and composition. If we were to compare the raw data instead, the variance of household income would be larger than that of male earnings, as seen in figure 1.

(43.) As already noted in section I, in the online appendix we present estimates of our nonstationary ECM, and the corresponding variance decompositions, for a sample of male labor earnings and total household income from the PSID. In the PSID samples, the transitory variance component appears to have played more of a role for both male earnings and total household income.

(44.) We analyze increasingly broad income aggregates, rather than individual income categories separately, because for many households, income from at least some of these individual categories is zero. The large number of zero-income observations makes it difficult to estimate the ECM separately for each income category.

(45.) We use our male-headed households sample so as not to confound the effects of moving to broader measures of income with the effects of moving to broader samples.

(46.) See, however, Piketty and Saez (2007), who find a decrease in progressivity between 1960 and 2004, which was driven primarily by changes in corporate taxes and in estate and gift taxes, which are not included in our analysis.

(47.) The inclusion of the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] component renders the estimation of the model more challenging. Indeed, we have found the estimation of this model to be much less numerically stable than that of our baseline ECM, and the estimates of the variance of the persistent innovations ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]) are very noisy. As in the case of our baseline ECM, we impose smoothness restrictions on the [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] series by restricting it to a fourth-degree polynomial, for the reasons discussed in section V.B. We thank Greg Kaplan for sharing computer code that helped with the estimation of this specification. Note also that in this specification the timing of the effects of changes in model parameters on changes in income inequality is different from that in our baseline model, because of the presence of the [[phi].sub.t] parameters (changes in the variance of persistent shocks). In particular, in the alternative ECM, changes in the variance of persistent shocks will have lagged effects on income inequality. To see this, suppose for simplicity that [psi] = 1, so that [p.sup.i.sub.a,t] is a random walk and the persistent shocks [[eta].sup.i.sub.a,t] accumulate over time. Next, suppose, for example, that the variance of persistent shocks experiences a one-time permanent increase in year t (there is a one-time permanent jump in [[phi].sub.t]). Then, over time, as new cohorts enter the adult (ages 25-60) population, they will face the larger persistent shocks, and these shocks accumulate over time. Therefore, the one-time permanent increase in the variance of persistent shocks in year t would continue to lead to increases in inequality in future periods, as younger cohorts (facing larger persistent shocks) replace the older cohorts (which have accumulated smaller persistent shocks over their lifetime). One implication of this is that, if the model in equations A. 1 through A.4 were the correct representation of the world (and especially if [psi] = 1), and if it were the case that the variance of persistent shocks had increased permanently some time before 1987 (the beginning of our sample), then part of the increase in income inequality after 1987 would be the result of the increase in the variance of the persistent shocks before 1987. Our baseline ECM would likely attribute such changes in inequality to [[lambda].sub.t]. We thank Greg Kaplan for making this observation.

(48.) According to this model specification (and our data), there has been no distinct trend in the variances of persistent or transitory shocks in our sample period. All the increase in the variance of the persistent component of earnings comes from an increase in the "price" of permanent characteristics. This is entirely consistent with our findings from our baseline ECM, where the rise in the variance comes from an increase in the price of permanent and persistent characteristics. One might ask, both in the context of this alternative model and in the context of our baseline ECM, to what extent this increase in the price of certain permanent or persistent characteristics represents increases in the returns to observable characteristics (such as education and experience) versus unobservable ones. The large causal literature on earnings and wage inequality in labor economics indicates that the answer is both, as it generally finds increases in inequality both between and within narrowly defined education and experience groups (see, for instance, Lemieux 2008).

(49.) We have also estimated the model using the identity matrix as weighting matrix. The results (not reported) are very similar.

(50.) When [[lambda].sub.t] is unrestricted, we use the normalization [[pi].sub.2008] = [[pi].sub.2009], since in that case [[lambda].sub.t] and [[pi].sub.t] cannot be identified separately in the last year of the sample, t = 2009. Results for the unrestricted version are presented in the online appendix.

Table 1. Descriptive Statistics for the Income Measures, 1987-2009 (a)

 Log of male earnings

 No. of
Year obs. Mean SD

1987 8,180 10.38 0.78
1988 8,670 10.36 0.81
1989 9,019 10.33 0.81
1990 9,081 10.33 0.81
1991 8,891 10.32 0.81
1992 8,899 10.32 0.83
1993 9,240 10.29 0.84
1994 9,354 10.30 0.83
1995 9,522 10.32 0.83
1996 9,498 10.33 0.83
1997 9,608 10.37 0.82
1998 9,806 10.39 0.83
1999 9,865 10.43 0.82
2000 9,933 10.45 0.82
2001 9,978 10.46 0.82
2002 9,946 10.45 0.84
2003 9,895 10.42 0.84
2004 9,980 10.43 0.84
2005 10,048 10.43 0.84
2006 10,317 10.43 0.85
2007 10,574 10.44 0.84
2008 10,505 10.42 0.85
2009 10,290 10.39 0.87

Total or 221,099 10.38 0.83
average

 Log of pre-tax household income

 Male-headed households All households

 No. of No. of
Year obs. Mean SD obs. Mean SD

1987 8,161 10.65 0.78 12,789 10.45 0.85
1988 8,643 10.65 0.81 13,217 10.45 0.87
1989 8,991 10.63 0.83 13,625 10.43 0.88
1990 9,048 10.63 0.82 13,871 10.42 0.88
1991 8,858 10.60 0.83 14,058 10.40 0.88
1992 8,875 10.62 0.84 14,227 10.40 0.90
1993 9,215 10.61 0.85 14,461 10.39 0.90
1994 9,339 10.62 0.85 14,669 10.39 0.90
1995 9,494 10.64 0.85 14,980 10.40 0.92
1996 9,466 10.66 0.86 14,931 10.42 0.93
1997 9,588 10.69 0.87 15,253 10.45 0.93
1998 9,784 10.72 0.88 15,626 10.49 0.94
1999 9,830 10.77 0.88 15,772 10.52 0.94
2000 9,896 10.79 0.88 15,956 10.53 0.95
2001 9,939 10.78 0.87 16,114 10.53 0.94
2002 9,905 10.77 0.88 16,155 10.53 0.93
2003 9,848 10.76 0.87 16,198 10.51 0.94
2004 9,920 10.76 0.89 16,339 10.52 0.96
2005 10,001 10.75 0.90 16,540 10.51 0.96
2006 10,272 10.77 0.92 16,944 10.52 0.97
2007 10,516 10.77 0.92 17,469 10.51 0.97
2008 10,468 10.74 0.90 17,427 10.49 0.95
2009 10,247 10.72 0.91 17,354 10.45 0.95

Total or 220,304 10.70 0.86 353,975 10.47 0.92
average

 Log of after-tax household income

 Male-headed households All households

 No. of No. of
Year obs. Mean SD obs. Mean SD

1987 8,155 10.48 0.73 12,783 10.29 0.80
1988 8,634 10.48 0.76 13,211 10.29 0.82
1989 8,982 10.46 0.78 13,616 10.27 0.82
1990 9,045 10.45 0.77 13,859 10.26 0.82
1991 8,849 10.43 0.77 14,045 10.25 0.82
1992 8,867 10.45 0.79 14,216 10.25 0.83
1993 9,209 10.44 0.79 14,457 10.24 0.84
1994 9,336 10.46 0.79 14,668 10.25 0.83
1995 9,487 10.47 0.79 14,987 10.26 0.84
1996 9,459 10.49 0.80 14,944 10.28 0.85
1997 9,582 10.52 0.80 15,252 10.31 0.85
1998 9,775 10.56 0.82 15,630 10.35 0.86
1999 9,826 10.60 0.82 15,773 10.38 0.86
2000 9,893 10.62 0.82 15,958 10.39 0.87
2001 9,936 10.62 0.81 16,117 10.39 0.86
2002 9,899 10.62 0.81 16,161 10.40 0.85
2003 9,843 10.62 0.82 16,198 10.39 0.86
2004 9,916 10.63 0.83 16,342 10.41 0.88
2005 9,995 10.62 0.84 16,541 10.40 0.88
2006 10,270 10.63 0.86 16,960 10.41 0.89
2007 10,511 10.63 0.87 17,474 10.40 0.90
2008 10,463 10.61 0.84 17,425 10.38 0.88
2009 10,241 10.59 0.84 17,355 10.36 0.87

Total or 220,173 10.54 0.81 353,972 10.33 0.85
average

Source: Authors' calculations using data from the Statistics of
Income Division (SOI) of the Internal Revenue Service.

(a.) See sections 11.13 and II.C in the text for definitions of the
income measures and of the samples, respectively. SD = standard
deviation.

Table 2. Estimated Linear Time Trends of Persistent and Transitory
Variance in Male Labor Earnings (a)

 Estimated component and
 decomposition method

 Persistent component

 Error
 KSS GM components
 method method model

Coefficient 0.0037 0.0037 0.0038
 on linear (0.0002) (0.0002) (0.0003)
 time
 trend

p value 0.000 0.000 0.000
[R.sup.2] 0.95 0.94 0.89

 Estimated component and
 decomposition method

 Transitory component

 Error
 KSS GM components
 method method model

Coefficient 0.0000 0.0001 0.0001
 on linear (0.0002) (0.0002) (0.0004)
 time
 trend

p value 0.947 0.610 0.746
[R.sup.2] 0.00 0.02 0.01

Source: Authors' regressions using SOI data.

(a.) Each column reports results of an ordinary least squares
regression of the persistent or the transitory component of the
variance in male labor earnings, as calculated by the indicated
decomposition method, on a constant (not reported) and a linear
trend. Standard errors are in parentheses.

Table 3. Estimates of Stationary Error Components Models,

 Income measure and
 sample

 Pre-tax household income

 Male labor Male-headed All
Parameter earnings households households

Unrestricted model

[[sigma].sup.2.sub.[alpha]] 0.1968 0.1885 0.1960
 (0.0018) (0.0018) (0.0016)
[psi] 0.9623 0.9717 0.9669
 (0.0010) (0.0012) (0.0007)
[[sigma].sup.2.sub.[eta]] 0.0293 0.0183 0.0269
 (0.0007) (0.0006) (0.0006)
[[sigma].sup.2.sub.[epsilon]] 0.1826 0.1405 0.1577
 (0.0034) (0.0038) (0.0032)
[[theta].sub.1] 0.2286 0.3072 0.2766
 (0.0144) (0.0191) (0.0148)
[[theta].sub.2] 0.1231 0.2131 0.1639
 (0.0151) (0.0206) (0.0154)

Restricted model ([psi = 1)

[[sigma].sup.2.sub.[alpha]] 0.2431 0.2162 0.2391
 (0.0014) (0.0014) (0.0013)
[[sigma].sup.2.sub.[eta]] 0.0093 0.0076 0.0095
 (0.0001) (0.0001) (0.0001)
[[sigma].sup.2.sub.[epsilon]] 0.2069 0.1512 0.1756
 (0.0035) (0.0040) (0.0033)
[[theta].sub.1] 0.3477 0.3830 0.3875
 (0.0116) (0.0168) (0.0127)
[[theta].sub.2] 0.2895 0.3276 0.3313
 (0.0145) (0.0207) (0.0160)

 Income measure and
 sample

 After-tax household
 income

 Male-headed All
Parameter households households

Unrestricted model

[[sigma].sup.2.sub.[alpha]] 0.1533 0.1579
 (0.0015) (0.0013)
[psi] 0.9805 0.9770
 (0.0011) (0.0007)
[[sigma].sup.2.sub.[eta]] 0.0135 0.0187
 (0.0005) (0.0004)
[[sigma].sup.2.sub.[epsilon]] 0.1199 0.1387
 (0.0031) (0.0026)
[[theta].sub.1] 0.3066 0.2772
 (0.0186) (0.0136)
[[theta].sub.2] 0.2185 0.1734
 (0.0203) (0.0142)

Restricted model ([psi = 1)

[[sigma].sup.2.sub.[alpha]] 0.1713 0.1854
 (0.0012) (0.0011)
[[sigma].sup.2.sub.[eta]] 0.0072 0.0089
 (0.0001) (0.0001)
[[sigma].sup.2.sub.[epsilon]] 0.1262 0.1492
 (0.0032) (0.0026)
[[theta].sub.1] 0.3608 0.3528
 (0.0168) (0.0119)
[[theta].sub.2] 0.2998 0.2852
 (0.0202) (0.0142)

Source: Authors' calculations using SOI data.

(a.) Estimates of equations 2 through 5 in the text using a minimum
distance estimator (see section V.Q. Asymptotic standard errors are
in parentheses.

Table 4. Estimates of Nonstationary Error Components Model (a)

 Income measure and
 sample

 Pre-tax household
 income

 Male labor Male-headed All
Parameter earnings households households

Persistent component

[[sigma].sup.2.sub.[alpha]] 0.1742 0.1566 0.1701
 (0.0027) (0.0027) (0.0022)
[psi] 0.9631 0.9751 0.9687
 (0.0010) (0.0012) (0.0008)
[[sigma].sup.2.sub.[eta]] 0.0246 0.0129 0.0209
 (0.0008) (0.0005) (0.0005)

[[lambda].sub.t] polynomial (b)
[b.sub.1] 0.0226 0.0275 0.0132
 (0.0048) (0.0055) (0.0039)
[b.sub.2] (x 10) -0.0273 -0.0198 0.0019
 (0.0089) (0.0103) (0.0071)
[b.sub.3] (x 100) 0.0151 0.0073 -0.0060
 (0.0061) (0.0071) (0.0049)
[b.sub.4] (x 1000) -0.0029 -0.0012 0.0016
 (0.0014) (0.0016) (0.0111)

Transitory component

[[theta].sub.1] 0.2343 0.3273 0.2905
 (0.0141) (0.0181) (0.0133)
[[theta].sub.2] 0.1262 0.2306 0.1762
 (0.0148) (0.0198) (0.0140)
[[sigma].sup.2.sub.[epsilon]] 0.1834 0.1354 0.1493
 (0.0119) (0.0101) (0.0100)
[[pi].sub.87] (c) 1.0000 1.0000 1.0000
[[pi].sub.88] 1.0792 1.0737 1.0642
 (0.0447) (0.0557) (0.0499)
[[pi].sub.89] 1.0352 1.0715 1.0493
 (0.0443) (0.0521) (0.0456)
[[pi].sub.90] 0.9763 0.9597 1.0015
 (0.0439) (0.0553) (0.0459)
[[pi].sub.91] 0.9611 0.9666 0.9853
 (0.0469) (0.0549) (0.0476)
[[pi].sub.92] 1.0266 1.0058 1.0141
 (0.0544) (0.0586) (0.0503)
[[pi].sub.93] 1.0342 0.9858 1.0130
 (0.0480) (0.0589) (0.0493)
[[pi].sub.94] 0.9657 0.9304 0.9573
 (0.0479) (0.0551) (0.0445)
[[pi].sub.95] 0.9925 0.9584 0.9997
 (0.0449) (0.0553) (0.0466)
[[pi].sub.96] 0.9798 0.9604 1.0039
 (0.0430) (0.0516) (0.0441)
[[pi].sub.97] 0.9628 1.0012 1.0126
 (0.0447) (0.0559) (0.0457)
[[pi].sub.98] 0.9684 1.0396 1.0584
 (0.0438) (0.0574) (0.0466)
[[pi].sub.99] 0.9548 1.0224 1.0226
 (0.0442) (0.0488) (0.0405)
[[pi].sub.00] 0.9785 1.0029 1.0217
 (0.0497) (0.0556) (0.0443)
[[pi].sub.01] 0.9665 0.9652 0.9760
 (0.0466) (0.0581) (0.0453)
[[pi].sub.02] 1.0284 1.0175 0.9769
 (0.0496) (0.0543) (0.0435)
[[pi].sub.03] 1.0155 0.9576 1.0044
 (0.0457) (0.0548) (0.0455)
[[pi].sub.04] 0.9909 1.0385 1.0872
 (0.0503) (0.0623) (0.0476)
[[pi].sub.05] 0.9810 1.0941 1.1010
 (0.0497) (0.0612) (0.0482)
[[pi].sub.06] 1.0379 1.1863 1.1457
 (0.0513) (0.0624) (0.0507)
[[pi].sub.07] 0.9854 1.1695 1.1512
 (0.0521) (0.0645) (0.0520)
[[pi].sub.08] 1.0335 1.0562 1.0522
 (0.0483) (0.0613) (0.0489)
[[pi].sub.09] 1.0763 1.0989 1.0555
 (0.0479) (0.0625) (0.0500)

 Income measure and
 sample

 After-tax household
 income

 Male-headed All
Parameter households households

Persistent component

[[sigma].sup.2.sub.[alpha]] 0.1328 0.1445
 (0.0024) (0.0020)
[psi] 0.9831 0.9784
 (0.0012) (0.0007)
[[sigma].sup.2.sub.[eta]] 0.0103 0.0158
 (0.0004) (0.0004)

[[lambda].sub.t] polynomial (b)
[b.sub.1] 0.0269 0.0126
 (0.0056) (0.0040)
[b.sub.2] (x 10) -0.0279 -0.0090
 (0.0106) (0.0074)
[b.sub.3] (x 100) 0.0143 0.0041
 (0.0073) (0.0050)
[b.sub.4] (x 1000) -0.0029 -0.0009
 (0.0017) (0.0011)

Transitory component

[[theta].sub.1] 0.3230 0.2880
 (0.0184) (0.0131)
[[theta].sub.2] 0.2316 0.1827
 (0.0202) (0.0139)
[[sigma].sup.2.sub.[epsilon]] 0.1123 0.1264
 (0.0089) (0.0086)
[[pi].sub.87] (c) 1.0000 1.0000
[[pi].sub.88] 1.0910 1.0815
 (0.0597) (0.0519)
[[pi].sub.89] 1.0911 1.0651
 (0.0559) (0.0475)
[[pi].sub.90] 0.9915 1.0242
 (0.0571) (0.0470)
[[pi].sub.91] 0.9676 0.9935
 (0.0576) (0.0489)
[[pi].sub.92] 1.0191 1.0318
 (0.0606) (0.0505)
[[pi].sub.93] 0.9989 1.0384
 (0.0608) (0.0495)
[[pi].sub.94] 0.9209 0.9432
 (0.0567) (0.0455)
[[pi].sub.95] 0.9273 0.9869
 (0.0572) (0.0475)
[[pi].sub.96] 0.9409 0.9974
 (0.0536) (0.0449)
[[pi].sub.97] 1.0140 1.0197
 (0.0578) (0.0475)
[[pi].sub.98] 1.0503 1.0764
 (0.0595) (0.0485)
[[pi].sub.99] 1.0430 1.0331
 (0.0510) (0.0419)
[[pi].sub.00] 1.0320 1.0374
 (0.0584) (0.0457)
[[pi].sub.01] 0.9653 0.9743
 (0.0605) (0.0461)
[[pi].sub.02] 1.0059 0.9835
 (0.0556) (0.0445)
[[pi].sub.03] 0.9712 1.0222
 (0.0569) (0.0471)
[[pi].sub.04] 1.0560 1.1077
 (0.0641) (0.0490)
[[pi].sub.05] 1.1173 1.1317
 (0.0642) (0.0493)
[[pi].sub.06] 1.2276 1.1971
 (0.0664) (0.0511)
[[pi].sub.07] 1.2034 1.1978
 (0.0688) (0.0534)
[[pi].sub.08] 1.0700 1.0815
 (0.0646) (0.0506)
[[pi].sub.09] 1.0909 1.0707
 (0.0641) (0.0510)

Source: Authors' calculations using SOI data.

(a.) Estimates of equations 6 through 9 in the text using a minimum
distance estimator (see section V.Q. Asymptotic standard errors are
in parentheses.

(b.) See appendix D for specification of the polynomial.

(c.) Parameters [[pi].sub.87] through [[pi].sub.09] correspond to the
years of the sample period (1987-09) and are normalized to equal 1 in
1987 (see appendix D).

Table 5. Estimated Linear Time Trends of Persistent and Transitory
Variance in Pre-Tax Household Income (a)

 Estimated component and decomposition
 method

 Persistent component

 KSS GM Error components
Sample method method model

Male-headed households

Coefficient on linear 0.0050 0.0048 0.0040
time trend variable (0.0003) (0.0003) (0.0005)

p value 0.000 0.000 0.000
[R.sup.2] 0.94 0.93 0.73

All households

Coefficient on linear 0.0056 0.0054 0.0048
time trend variable (0.0004) (0.0004) (0.0005)

p value 0.000 0.000 0.000
[R.sup.2] 0.94 0.93 0.80

 Estimated component and decomposition
 method

 Transitory component

 KSS GM Error components
Sample method method model

Male-headed households

Coefficient on linear 0.0007 0.0009 0.0016
time trend variable (0.0001) (0.0001) (0.0006)

p value 0.000 0.000 0.010
[R.sup.2] 0.78 0.92 0.28

All households

Coefficient on linear 0.0008 0.0010 0.0013
time trend variable (0.0001) (0.0001) (0.0005)

p value 0.000 0.000 0.010
[R.sup.2] 0.77 0.85 0.28

Source: Authors' regressions using SOI data.

(a.) Each column in each panel reports results of an ordinary least
squares regression of the persistent or the transitory component of
the variance in household income, as calculated by the indicated
decomposition method, on a constant (not reported) and a linear time
trend. Standard errors are in parentheses.