Rising Inequality: Transitory or Permanent? New Evidence from a U.S. Panel of Household Income 1987-2006

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Rising Inequality: Transitory or Permanent? New Evidence from a U.S. Panel of Household Income 1987-2006 Jason DeBacker, Bradley Heim, Vasia Panousi, and Ivan Vidangos 2011-60 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Rising Inequality: Transitory or Permanent? New Evidence from a U.S. Panel of Household Income 1987-2006∗ Jason DeBacker (Treasury Department) Bradley Heim (Indiana University) Vasia Panousi (Federal Reserve Board) Ivan Vidangos (Federal Reserve Board) December 15, 2011 Abstract We use a new and large panel dataset of household income to shed light on the permanent versus transitory nature of rising inequality in individual male labor earnings and in total household income, both before and after taxes, in the United States over the period 1987-2006. Due to the quality and the signi(cid:28)cant size of our dataset, we are able to conduct our analysis using rich and precisely estimated error-components models of income dynamics. Our main speci(cid:28)cation (cid:28)nds evidence for a quadratic heterogeneous income pro(cid:28)les component and a random walk component in permanent earnings, and foramoving-averagecomponentinautoregressivetransitoryearnings. We(cid:28)ndthatthe increase in inequality over our sample period was entirely permanent for male earnings, andpredominantlypermanentforhouseholdincome. Wealsoshowthatthetaxsystem, though reducing inequality, nonetheless did not materially a(cid:27)ect its increasing trend. Furthermore, wecompareourmodel-based(cid:28)ndingsagainstthoseofsimpler, non-model based inequality decomposition methods. We show that the results for the trends in the evolution of the permanent and transitory variances are remarkably similar across methods, whereastheresultsforthesharesofthosevariancesincross-sectionalinequalitydi(cid:27)erwidely. Furtherinvestigationintothesourcesofthesedi(cid:27)erencessuggeststhat simpler methods produce erroneous decompositions because they cannot (cid:29)exibly capture the relative degree of persistence of the transitory component of income. JEL codes: C33, D31, J31 ∗This paper has also been circulated as "Rising Inequality: Transitory or Permanent? New Evidence from a Panel of U.S. Tax Returns 1987-2006". Jason.DeBacker@treasury.gov, heimb@indiana.edu, vasia.panousi@frb.gov, ivan.vidangos@frb.gov. We are grateful to Joe Altonji, Chris Carroll, Eric Engen, Michael Golosov, Michael Palumbo, Dimitris Papanikolaou, Emmanuel Saez, Dan Sichel, and Paul Smith for helpful comments and suggestions. We also thank seminar participants at the Federal Reserve Board, the NBER Summer Institute Consumption group and Labor Studies group, the Federal Reserve Bank of San Francisco, Colegio de Mexico, Bank of Mexico, Bank of Portugal, Applied Micro System Conference at the Federal Reserve Bank of St Louis, the meetings of the European Economic Association and the European Association of Labor Economists, and the U.S. Census Bureau for very constructive discussions. 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 sta(cid:27)s. 1

1 Introduction An extensive literature has documented a large increase in income inequality in the United States in recent decades. In this paper, we ask whether this observed increase in crosssectional inequality re(cid:29)ected an increase in permanent or in transitory inequality. By permanent, we mean an increase in long-run inequality, or growing dispersion in permanent incomes. By transitory, we mean greater short-run variability in incomes, or individuals moving around more within the income distribution at relatively short frequencies of one to a few years. The distinction between permanent and transitory inequality is important for two main reasons. First, it is useful in evaluating the proposed explanations for the documented increase in annual cross-sectional inequality. For example, if rising inequality re(cid:29)ects solely an increase in permanent inequality, then consistent explanations would include, for example, skill-biased technical change. By contrast, an increase in transitory inequality could be due to increases in income mobility, perhaps driven by growing job instability for all workers. Second, the distinction is useful because it informs the welfare evaluation of cross-sectional inequality increases. Speci(cid:28)cally, lifetime income is a measure of long-term available resources, and hence an increase in permanent inequality would be welfare-reducing according to most social welfare functions. By contrast, increasing transitory inequality would have less of an e(cid:27)ect on welfare, especially in the absence of liquidity constraints restricting consumption smoothing. Furthermore, as the next section outlines, the literature has not as yet reached a clear consensus about either the nature or the timing of the increases in each inequality component in the last two decades. One important aspect of our contribution is the use of a new and superior data source to shed new light on the permanent-vs-transitory decomposition of inequality and its evolution over time. In particular, we use a large panel of household income from tax returns to study the permanent-vs-transitory nature of rising inequality in individual (male) labor earnings and in total household income, both before and after taxes, in the United States over the 1 period 1987-2006. Our panel constitutes a one-in-5,000 random sample of the population of U.S. taxpayers. It contains individual-level labor earnings information as well as householdlevel income information reported in tax forms. It also includes information on the age and gender of the primary and secondary tax (cid:28)lers from matched social security records. Our broadest sample consists of nearly 300,000 observations on 30,000 households, and is 1 The data are kept at the Treasury Department, and all of our analysis was run at and cleared by the Treasury, in order to maintain the con(cid:28)dentiality of the data. 2

therefore substantially larger than the typical survey panels, such as the Panel Study of Income Dynamics (PSID), used to address related questions in the literature. In addition, our data are not subject to top-coding, and are less likely to be a(cid:27)ected by measurement error, compared to survey data. The quality and size of our dataset allow us to start the analysis by precisely estimating rich error-components models of income dynamics. In particular, error-components models fully specify the process that generates income over time, and can be used to decompose the cross-sectional variance of log income(cid:22)our measure of inequality(cid:22)into permanent and transitory parts. This way, we can explore in detail the role of permanent and transitory income components for the evolution of inequality. Indeed, one of the main advantages of suchmodelsisthattheyaresu(cid:30)cientlydetailedand(cid:29)exibletobeabletocapturemanyfacets both of the autocorrelation of earnings and of the evolution of earnings over the lifecycle. We therefore view the fact that we can use the large size of our data to ensure that our models are very precisely estimated as one of the main contributions of our paper. We next expand our analysis to explore simpler, approximate decomposition methods that have been used in the literature to examine the permanent-vs-transitory nature of inequality. Here, one important aspect of our contribution is that we investigate the relation across the di(cid:27)erent inequality-decomposition methods, model-based as well as non-modelbased, and we propose an explanation for the di(cid:27)erences across methods. This is important because, in the existing literature, it is impossible to discern whether the di(cid:27)erences across studies are due to the di(cid:27)erent data or to the di(cid:27)erent methodologies used. By contrast, we clarify the connections across di(cid:27)erent methods and we propose one way of thinking about the variety of results they yield. Hence, our analysis could provide guidance for researchers as to the potential outcomes of choosing between di(cid:27)erent methods. Turning to the details of our error-components models, our data indicate that male labor earnings are best described by a combination of permanent and transitory components with the following features. First, permanent earnings, whose relative importance varies over calendar time, are captured by the sum of a quadratic heterogeneous income pro(cid:28)les component and a random walk component. Second, transitory earnings are characterized by an ARMA(1,1) process with year-speci(cid:28)c innovation variances. Transitory earnings turn out to be relatively persistent (the autoregressive coe(cid:30)cient in our baseline speci(cid:28)cation is 0.63), despite the inclusion of both heterogeneous income pro(cid:28)les and a random walk in permanent earnings. For household income, the main di(cid:27)erence is that the data provide less support for the inclusion of a random walk component in permanent income. 3

Our main (cid:28)ndings on the permanent-vs-transitory decomposition of inequality are as follows. For male labor earnings, we (cid:28)nd that the entire increase in cross-sectional inequality overthe1987-2006periodwaspermanent. Inparticular, we(cid:28)ndthatthepermanentvariance of (log) male earnings increased over this period, while the transitory variance did not. In terms of the permanent-vs-transitory makeup of the cross-sectional variance of male earnings at a single point in time, our preferred model speci(cid:28)cation implies that, on average, 65% of the total variance of male labor earnings was permanent, while 35% was transitory. For total household income, we (cid:28)nd that the large increase in inequality over our sample periodwaspredominantly, thoughnotentirely, permanent. Forthisbroaderincomecategory, both the permanent and the transitory parts of the cross-sectional variance increased, with the permanent variance contributing two thirds and the transitory variance one third of the increase in the total cross-sectional variance. Furthermore, the increase in the transitory component re(cid:29)ected an increase in the transitory variance of spousal labor earnings, transfer income, and investment income. Weextensivelyexplorethesensitivityofouranalysistomodelspeci(cid:28)cation. Weshowthat the results for the trends in the permanent and transitory variances are remarkably robust across model speci(cid:28)cations. However, the shares of cross-sectional inequality attributed to the permanent and transitory income components at a given point in time are quite sensitive to the speci(cid:28)cation. For instance, the transitory share of total cross-sectional inequality ranges from 27% to 60%, depending on the model used. We examine the reasons for these di(cid:27)erences and we show that they re(cid:29)ect di(cid:27)erences in the (relative) degree of persistence in the transitory earnings process across the various model speci(cid:28)cations. Intuitively, when transitory earnings are less persistent, then more of the persistence in the earnings data will be attributed to the permanent component, leading to a larger role assigned to permanent earnings overall. Conversely, when transitory earnings are allowed to be more persistent, then more of the persistence in the data will be picked up by the transitory part, leading to a larger role played by the transitory earnings component. Turning to the comparison with simpler methods, we look in particular into two approximate methods that decompose the cross-sectional variance into permanent and transitory parts without explicitly using error-components models. These methods essentially de(cid:28)ne permanent income as the average of annual income over a certain period of time, and then transitory income as the deviations of annual income from that average. Here, we (cid:28)nd that the results for the trends of the permanent and transitory variance components are surprisingly robust across methods, and that therefore the approximate methods corroborate 4

2 our model-based results for the inequality trends. However, the permanent and transitory shares of the cross-sectional variance turn out to be very sensitive to the method used. We then propose that the reasons for these di(cid:27)erences are closely related to the reasons for the di(cid:27)erences across the di(cid:27)erent model speci(cid:28)cations. In particular, the approximate methods we consider do not allow for persistence in transitory income. As a result, they attribute to the permanent income component part of what is in reality a transitory (though serially correlated) shock, thereby overstating the importance of the permanent part of inequality. In other words, the simpler decompositions rely on restrictions that are strongly rejected by the data. Clearly, the share of cross-sectional inequality attributed to the permanent versus the transitory component will have important quantitative and policy implications for the welfare costs of income inequality. Therefore, our analysis provides signi(cid:28)cant guidance in that direction and it indicates that the search for the appropriate inequality-decomposition method needs to carefully consider the nature of the data, with particular emphasis on the relative degree of persistence in the transitory component of income. Finally, our tax return data also allow us to examine in detail the role of the federal tax system for the evolution of income inequality. In particular, we investigate whether the evolution of inequality for after-tax household income di(cid:27)ers materially from the evolution of inequality for pre-tax income. Our measure of after-tax household income re(cid:29)ects all federal personal income taxes, including all refundable tax credits, as well as payroll taxes. We (cid:28)nd that the cross-sectional variance of after-tax income is on average 15% smaller than the variance of pre-tax income, re(cid:29)ecting the overall progressivity of the U.S. federal tax system. On net, however, the e(cid:27)ect of the tax system in reducing income inequality appears quite stable over the sample period. In other words, the tax system does not appear to have signi(cid:28)cantly altered the trend toward rising inequality, despite the large changes in tax policy over this period. The rest of the paper is organized as follows. The next section discusses the related literature and places our results in the context of existing studies. Section 3 describes our dataset, our sample selection, and the trends in income inequality in our data. Section 4 introduces our error-components models and discusses their estimation. Section 5 presents the model estimates for male labor earnings and uses the estimated models to decompose the cross-sectional variance into permanent and transitory parts. The section then examines the sensitivity of the results to model speci(cid:28)cation. Section 6 compares our model results to those from alternative methods of analysis and discusses the reasons for the di(cid:27)erences 2 We also examine the evolution of measures of dispersion in the distribution of income changes over one and two years (volatility), which provide yet further support for our model-based inequality-trend (cid:28)ndings. 5

across methods. Section 7 examines total household income and the role of the U.S. federal tax system for the evolution of income inequality. Section 8 describes several robustness tests, and section 9 concludes. Technical details and additional results are provided in the Appendix. 2 Related Literature An extensive literature has documented a large increase in labor earnings inequality in the 3 U.S.inrecentdecades. Asmallbranchoftheliteraturehasattemptedtodeterminewhether this increase in cross-sectional inequality re(cid:29)ected an increase in permanent or in transitory inequality. The earlier studies, including Gottschalk and Mo(cid:30)tt (1994); Mo(cid:30)tt and Gottschalk (1995); and Haider (2001), all use PSID data, and generally conclude that a substantial part (as much as one half) of the increase in cross-sectional inequality in the 1970s and early 1980s was transitory. There are very few studies analyzing the last two decades, although earnings inequality has continued to increase. Furthermore, the results across studies are not conclusive. For example, using the PSID, Gottschalk and Mo(cid:30)tt (2008) (cid:28)nd that the transitory variance has not increased after the mid-to-late 1980s, whereas Heathcote, Perri, and Violante (2010) conclude that the transitory variance rose substantially in the 1990s. Kopczuk, Saez, and Song (2010), using Social Security earnings data, (cid:28)nd that the increase in inequality from the 1970s to the early 2000s was entirely permanent. However, they use only a simple approximate decomposition method and their (cid:28)ndings contradict the more-established results of the earlier literature for the 1970s and early 1980s, raising doubts about the factors driving their analysis for the more recent period as well. In this paper, we conclude that the increase in inequality since the mid-to-late 1980s has been entirely permanent. Furthermore, we con(cid:28)rm this (cid:28)nding with a variety of model speci(cid:28)cations, as well as di(cid:27)erent decomposition methods, thereby obtaining remarkably robust results. Inequality in total household income has also increased in recent decades, as documented, among others, by Krueger and Perri (2006); and Heathcote, Perri, and Violante (2010). The only studies that have in some way attempted to decompose the increase in household income inequality into permanent and transitory parts are Gottschalk and Mo(cid:30)tt (2009); Primiceri and van Rens (2009); and Blundell, Pistaferri, and Preston (2008). Gottschalk 3 For instance, Kopczuk, Saez, and Song (2010) use longitudinal earnings data from Social Security Administration (SSA) records to document that annual earnings inequality has increased steadily since the early 1950s. 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). 6

and Mo(cid:30)tt (2009) use simply an approximate method and provide only suggestive evidence of an increase in the transitory variance starting in the mid-1980s, without conducting a full analysis. By contrast, Primiceri and van Rens (2009), utilizing repeated cross-sections on income and consumption from the Consumer Expenditure Survey (CEX), (cid:28)nd that all of the increase in household income inequality in the 1980s and 1990s re(cid:29)ected an increase in the permanent variance. Our results indicate that, for the increase in the cross-sectional variance of household income, the transitory variance did play some role, though not as prominently 4 as Gottschalk and Mo(cid:30)tt (2009) seem to suggest. Furthermore, we show that the increase in the transitory variance of household income re(cid:29)ected an increase in the transitory variance of spousal labor earnings, transfer income, and investment income. Our paper is also related to a recent literature that has analyzed the trends in the dispersion of short-term income changes, or income volatility. The (cid:28)ndings in this literature have been more consistent across di(cid:27)erent studies. For instance, a 2008 study by the Congressional Budget O(cid:30)ce (CBO); Sabelhaus and Song (2009); Celik, Juhn, McCue, and Thompson (2011); and Shin and Solon (2011), all (cid:28)nd that the volatility of male earnings 5 did not increase between the 1980s and the early 2000s. Our male labor earnings data are consistent with the (cid:28)ndings in this literature, as we document no increase in male earnings volatility. However, we do (cid:28)nd an increase in the volatility of total household income. Finally, ourpaperalsocontributestoaliteraturethatmodelsandestimatesthedynamics of labor income. In particular, there is a long standing debate in this literature concerning the nature of individual labor earnings processes. According to one view, individuals face similar lifecycle earnings pro(cid:28)les and are subject to highly persistent (permanent) earnings shocks, so that permanent earnings are best re(cid:29)ected by a random walk component. See, for example, Hryshko (2010) for a recent study favoring this speci(cid:28)cation. According to the second view, individuals face heterogeneous lifecycle earnings pro(cid:28)les and are subject to less persistent shocks, so that permanent earnings are best captured by a (cid:16)heterogeneous income pro(cid:28)les(cid:17)component. See, for instance, the papers by Baker (1997); and Guvenen 6 (2009). Our baseline model nests both speci(cid:28)cations. For (male) labor earnings, we (cid:28)nd that the data support the inclusion of a heterogeneous income pro(cid:28)les component as well 4 Blundell,Pistaferri,andPreston(2008)(cid:28)ndalargeincreaseinthevarianceofpermanentincomeshocks intheearly1980s,followedbyalargeincreaseinthevarianceoftransitoryshocksinthelate1980s. However, we cannot directly compare our results with theirs, as our sample periods barely overlap. 5 Dynan, Elmendorf, and Sichel (2007) (cid:28)nd 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. 6 Guvenen (2007) investigates the di(cid:27)erences in the implications of these two speci(cid:28)cations of the labor income process for lifecycle consumption behavior. 7

as a random walk component. Furthermore, our data support a quadratic speci(cid:28)cation of heterogeneous income pro(cid:28)les. This is because, in our data, the lifecycle pro(cid:28)le of the crosssectional variance, controlling for either year or cohort e(cid:27)ects, is concave in the earlier part of the lifecycle, and convex in the later part. The heterogeneous income pro(cid:28)les component in our baseline model implies that the lifecycle pro(cid:28)le of the variance is a cubic polynomial in age, which (cid:28)ts the pro(cid:28)le in our tax data well. Overall, we (cid:28)nd stronger support for the (quadratic) heterogenous income pro(cid:28)les than for the random walk: The restriction that the variance of the innovation of the random walk component is zero is rejected at a 5% level but not at a 1% level. For total household income, we (cid:28)nd less support for a random walk component, whose inclusion depends on the speci(cid:28)c sample, on the level of the minimum threshold used to exclude low-income observations, and on whether the income data are before of after taxes. 3 Data This section describes our panel of income tax returns, our sample selection, and the income inequality trends observed in our data over the period 1987-2006. 3.1 Panel and Variable Description We use data from a twenty-year panel of tax returns spanning the period 1987-2006. To create this panel, we merged returns from an existing 1987-1996 Statistics of Income (SOI) panel, kept at the Treasury, with returns from cross-sectional (cid:28)les from 1997-2006. We then cut the sample to returns for which the primary (cid:28)ler had a social security number ending in one of two four-digit combinations. The resulting panel (with two exceptions noted below) 7 is a one-in-5,000 random sample of tax units followed over 1987-2006. Each of the data sources is next described in turn. The 1987-1996 panel was collected by the SOI and is kept at the Treasury Department. The panel started with a strati(cid:28)ed random sample of taxpayers who (cid:28)led in 1987, a subset of which was chosen based on the primary (cid:28)ler’s social security number ending in one of 7 Our sample is representative of the U.S. tax (cid:28)ling population. The fraction of U.S. households (cid:28)ling tax returns is generally around 90-95%, see for example Piketty and Saez (2003). Most households who do not (cid:28)le 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 (cid:28)rst-order concern, because, as documented by Autor, Katz, and Kearney (2008); and Kopczuk, Saez, and Song (2010), changes in income inequality in the U.S. over our sample period have been concentrated on the upper part of the income distribution. 8

two four-digit combinations. 8,9 All individuals represented on the tax return of a member of this cross section, including secondary taxpayers on joint returns and dependents, were considered to be members of the panel. Over the following nine years, the SOI division included in the panel all returns that reported any panel member as a primary or secondary taxpayer, including tax returns (cid:28)led by panel members who were dependents of another taxpayer. To keep the sample representative of the tax (cid:28)ling population in subsequent years, tax returns from tax years 1988 through 1996 were added to the panel if the primary (cid:28)ler had an social security number ending in one of the two aforementioned four-digit combinations 10 but did not (cid:28)le a return in 1987. In addition to information from each tax form, the dataset includes information on age and gender of the primary and secondary (cid:28)lers obtained from matched social security records. The 1997-2006 data come from yearly cross-sections collected by the SOI, and also maintained at the Treasury. Like the 1987 sample described above, a strati(cid:28)ed random sample was collected in each of these years, consisting partly of a strictly random sample based on the last four digits of the primary (cid:28)ler’s social security number. Each cross-section contains information from tax forms, and merged information on age and gender of the primary and secondary (cid:28)lers from social security records. As noted above, in our estimation sample we only include returns from either of these two data sources where the primary taxpayer’s social security number had one of the two 1987 original four-digit endings, resulting in a one-in-5,000 random sample. The panel is not balanced, as some taxpayers drop out of the sample due to death, emigration, or falling below the tax (cid:28)ling thresholds, while others enter because of immigration or becoming (cid:28)lers. The ideal measure of individual-level earnings for this study is gross labor income before any amounts are deducted for health insurance premiums or retirement account contributions. However, our data do not contain such a variable, and hence we use a measure of labor income that is as close to gross labor income as is possible, using tax data. For this, we take taxable wages, and we add reported contributions to retirement savings accounts. This measure of labor income will include all income that a taxpayer’s employer has reported, namely wages, salaries, and tips, as well as the portion of these that is placed in a retirement 8 The full 1987 strati(cid:28)ed random sample actually consisted of two parts, the random sample noted in the textandahigh-incomeoversample. Wedonotusethehigh-incomeoversampleinouranalysisinthispaper. 9 On tax returns in which a married couple is (cid:28)ling jointly, the primary (cid:28)ler is the individual listed (cid:28)rst on a tax form. This is usually, though not always, the husband. On tax returns of single (cid:28)lers, the primary (cid:28)ler is the individual who (cid:28)led the return. 10 However, taxpayers with one of the two social security number endings who (cid:28)led as dependents in 1987, or who were listed as a dependent or secondary (cid:28)ler in 1987, were not included in the sample. We discuss this issue in section 3.2. 9

account. Since our data do not include information on the health insurance premiums paid by the taxpayer and excluded from taxable wages, our measure of labor income will exclude those amounts. Our measure also excludes any income earned from self-employment. For pre-tax total household income, we start with the (cid:16)total income(cid:17) amount of income reported. This variable includes wages and salaries, dividends, alimony, business income, income from rental real estate, royalties, and trusts, unemployment compensation, capital gains, and taxable amounts of interest, IRA distributions, pensions, and social security bene(cid:28)ts. To this, we add back nontaxable interest, IRA distributions, pensions, and social security bene(cid:28)ts. There is some debate as to whether capital gains should be included in the measure of household income, as the amount of capital gains realized in a particular year and reported on the tax form may include gains that accrued in the past. Hence, it may make household income appear (cid:16)lumpier(cid:17) than it actually is, since income will be higher in years when gains from prior years are realized, and lower in years when gains accrued but were not realized. However, excluding capital gains will result in the measure of household income being too low for any taxpayer who had gains in that year (whether or not they were realized), and this downward bias will be quite large for taxpayers whose primary source of income is from investments. On balance, we feel that this concern is more important, and therefore we 11 include capital gains in our benchmark measure of household income. For after-tax household income, we start with the measure of pre-tax household income described above. We then subtract the amount of (cid:16)total taxes(cid:17) which captures total income taxes (including self-employment taxes) after non-refundable tax credits are taken into account. Next, we subtract the total amount of FICA taxes owed on the earned income of the couple. This is done to ensure that all federal taxes (including income and payroll taxes) are included for all taxpayers, regardless of whether they are wage and salary workers or self-employed. Finally, we add refundable tax credits (including the earned income tax credit and the refundable portion of the child tax credit) to arrive at our measure of after-tax household income. 3.2 Sampling Change and Demographics There was a change in the sampling frame of our data in 1996. As a result of this change, we are missing two groups of (cid:28)lers in the pre-1996 period: Dependent (cid:28)lers in 1987 over the 11 However,wehaveveri(cid:28)edtherobustnessofourresultstotheexclusionofcapitalgainsfromourmeasure of household income. 10

period 1987-1996, and non-dependent primary (cid:28)lers in 1988-1996 who were either dependent or secondary (cid:28)lers in 1987. These two groups primarily consist of young (in the case of dependents) or female (in the case of secondary) taxpayers. The e(cid:27)ect of missing these returns is therefore likely to be very small when we examine the labor income of males in their earning years, though it may be larger when we examine household income. To address potential issues introduced by this sampling change, we carry out our analysis of household income using two alternative samples. First, we analyze household income using the same sample of households that we use to analyze male earnings, namely maleheadedhouseholds,asthissamplewasessentiallyuna(cid:27)ectedbythesamplingchange. Second, we analyze household income using a sample with either a male or a female primary (cid:28)ler (see section 3.3). We are interested in this broader sample because it represents the entire population of tax units in the U.S., and not just those with a male primary (cid:28)ler. Oneadditionalpointtobementionedisthatourtaxdatacontainsfewersocio-demographic variables, compared to surveys like the PSID. Most importantly, though we have information on age and gender of the primary and secondary (cid:28)ler, we do not have information on education and race. We also lack information on hours of work, and hence our analysis will focus on annual earnings, as opposed to wage rates. 3.3 Sample Selection For the case of individual earnings, we restrict our sample to male primary (cid:28)lers, as is standard in the literature, because female movements in and out of the labor force introduce discontinuities in the earnings process. For household income, we carry out our analysis using two alternative samples. The (cid:28)rst sample includes households with a male primary (cid:28)ler only. This avoids confounding the e(cid:27)ects of using a broader measure of income (total household income) with the e(cid:27)ects of using a broader sample of households. In addition, this samplewasnota(cid:27)ectedbythechangeinsamplingframediscussedinsection3.2. Thesecond sample includes households with either a male or a female primary (cid:28)ler, and is representative of the population of U.S. taxpayers. For both male earnings and household income, we restrict our sample to households with a primary (cid:28)ler aged between 25 and 60. We impose this restriction because individuals in this age group are likely to have completed most of their formal schooling and are su(cid:30)ciently young not to be too strongly a(cid:27)ected by early retirement. For both male earnings and household income, we exclude earnings/income observations below a minimum threshold. For male earnings, since tax records do not provide information 11

on employment status or hours of work, we can exclude individuals with presumably weak labor force attachment only by dropping low-earnings observations. Turning to household income, we note that households with su(cid:30)ciently low income are not required to (cid:28)le taxes, although many actually do, so as to claim refundable tax credits, such as the earned income tax credit. In order to treat low-income observations consistently, we exclude observations 12 with reported household income below a minimum threshold. We take the relevant threshold to be one fourth of a full year full time minimum wage in 2004 ($2,575 in 2004), and 13 indexed for other years by nominal average wage growth. After imposing the restrictions above, we end up with a male earnings sample of 189,424 person-year observations. We refer to this sample as our ‘male earnings’ sample. We use this sample to analyze not only male earnings, but also household income, both before and after taxes. Our broader sample for household income, which includes households with either a male or a female primary (cid:28)ler, contains 294,910 observations. We refer to this sample as our ‘full’ household sample. Table I shows the number of observations, the mean, and the standard deviation for male earnings, pre-tax household income, and after-tax household income for each one of our samples. 3.4 Income Inequality Trends 1987-2006 We begin by documenting the evolution of inequality over time for male earnings and for household income, before and after taxes, in our panel of tax returns. Figures I (a) and I (b) show the cross-sectional variance (of the log) and the Gini coe(cid:30)cient, respectively, for male earnings, pre-tax household income, and after-tax household income. The (cid:28)gures show an increase in the variance and in the Gini coe(cid:30)cient for all three measures of income over 1987-2006. For example, the cross-sectional variance (of the log) increases by 0.11 points or 18% for male earnings, by 33% for pre-tax household income, and by 28% for after-tax 14 household income. In general, inequality in individual earnings is lower than inequality in household income. Furthermore, inequality in after-tax household income is lower than inequality in pre-tax household income, re(cid:29)ecting the progressivity of the U.S. tax system. Theseinequalitytrendsinourdataareconsistentwithtrendsthathavebeendocumented 12 In addition, it is well known that changes in income at low levels of income can unduly a(cid:27)ect model estimates. Two commonly used approaches to address this issue are to either exclude or to left-censor low-income observations. Given the issues discussed above, we choose to exclude them. 13 This threshold has also been used by Kopczuk, Saez, and Song (2010). In section 8 we check the sensitivity of our results to setting a lower/higher minimum threshold. 14 For household income, the (cid:28)gures use our ‘full’ household sample. In our male earnings sample, the cross-sectionalvariance(ofthelog)increasesbyabout40%forbothpre-taxandafter-taxhouseholdincome. 12

in many other U.S. studies. In the remainder of the paper, we focus on the cross-sectional variance (of the log) of earnings and household income as our measure of inequality, and we investigate whether the increase in the variance shown here represented an increase in permanent inequality versus an increase in transitory inequality. 4 Error-Components Models Our baseline model is as follows. Let yi denote log income, where i indexes individual, a a,t age, and t calendar year. 15 Log income is given by: yi = g(ζ ;Xi )+ξi , (1) a,t t a,t a,t where Xi is a vector of observable characteristics, g(.) is the part of log income that is a,t common to all individuals conditional on Xi , ζ is a vector of parameters (possibly including a,t t parameters that depend on calendar year t), and ξi is the unobservable error term. As is a,t common in the literature on income dynamics, we will remove the income variation that is due to observables, Xi , and focus on the dynamics of the error term, ξi . a,t a,t We model ξi as consisting of a permanent and a transitory component: a,t ξi = λ · (αi +βia+γia2 +ri )+ zi , where (2) a,t t a,t a,t (cid:124) (cid:123)(cid:122) (cid:125) (cid:124)(cid:123)(cid:122)(cid:125) permanent transitory ri = ri +(cid:15)i (3) a,t a−1,t−1 a,t zi = ρzi +π · ηi +θ · π · ηi (4) a,t a−1,t−1 t a,t t−1 a−1,t−1 αi ∼ iid(0,σ2), βi ∼ iid(0,σ2), γi ∼ iid(0,σ2), (5) α β γ cov(αi,βi) = σ , cov(αi,γi) = σ , cov(βi,γi) = σ , αβ αγ βγ (cid:15)i ∼ iid(0,σ2), ηi ∼ iid(0,σ2) a,t r a,t z The permanent income part consists of an individual-speci(cid:28)c, time-invariant component, αi, a quadratic heterogeneous income pro(cid:28)les component, βia + γia2, and a random-walk 15 The index a is actually (cid:16)normalized age(cid:17) or (cid:16)potential experience(cid:17), de(cid:28)ned as a = age−25, or years elapsed since age 25. 13

component, ri . These components are pre-multiplied by the year-speci(cid:28)c factor loading, a,t λ , which allows the relative importance of permanent income to vary over calendar time. t The components αi, βi, and γi are allowed to be freely correlated, with cov(αi,βi) = σ , αβ cov(αi,γi) = σ , and cov(βi,γi) = σ . αγ βγ When only allowing for a linear heterogeneous income component, we (cid:28)nd that the data strongly reject that speci(cid:28)cation. Allowing for a quadratic heterogeneous income pro(cid:28)les component improves the (cid:28)t of the model, and the quadratic speci(cid:28)cation cannot be rejected, evenifarandomwalkcomponentisalsoincluded. Inparticular, thequadraticheterogeneous income pro(cid:28)les component improves the (cid:28)t of the evolution of the cross-sectional variance of earnings/income over the lifecycle. This is because, in our data, the lifecycle pro(cid:28)le of the cross-sectional variance, controlling for either year or cohort e(cid:27)ects, is concave in the earlier part of the lifecycle, and convex in the later part. The heterogeneous income pro(cid:28)les componentinourmodelimpliesthatthelifecyclepro(cid:28)leofthevarianceisacubicpolynomial 16 in age, which (cid:28)ts the pro(cid:28)le in our tax data well. The transitory component in the model, zi , is speci(cid:28)ed as an ARMA(1,1) process. a,t The transitory innovations, ηi , are multiplied by the year-speci(cid:28)c factor loadings, π , which a,t t allow the variance of the innovations, and hence the relative importance of the transitory component, to vary by calendar year. Noticethat, inthemodelabove, permanentincomeshocks, (cid:15)i , arede(cid:28)nedasshocksthat a,t shift the path of income permanently, whereas transitory shocks, ηi , are de(cid:28)ned as shocks a,t with e(cid:27)ects that eventually disappear. Nonetheless, since transitory shocks are allowed to be serially correlated, it could take several years for their e(cid:27)ects to die out. In other words, the permanent component is de(cid:28)ned as capturing shocks that are not mean-reverting, whereas the transitory component is de(cid:28)ned as capturing mean-reverting shocks. For purposes of robustness, we are also interested in exploring how our decomposition of the cross-sectional variance into permanent and transitory parts depends on model speci(cid:28)cation. Therefore, we also examine three alternative models, which we call restricted models RM1, RM2, and RM3. These models are obtained by imposing the following restrictions on our baseline model: (i) RM1: σ2 = σ2 = σ = σ = σ = 0 (no heterogeneous income pro(cid:28)les) β γ αβ αγ βγ (ii) RM2: σ2 = 0 (no random walk) r (iii) RM3: θ = 0 (no MA transitory errors) In what follows we will demonstrate that the data strongly reject the restrictions imposed by 16 The evidence for heterogenous income pro(cid:28)les agrees with the (cid:28)ndings of Baker (1997) and Guvenen (2009) on PSID data, and of Baker and Solon (2003) on Canadian tax data. 14

all three models RM1, RM2, and RM3, thereby establishing our preference for our baseline model. 4.1 Estimation Estimation of our error-components models proceeds in two stages. In the (cid:28)rst stage, we construct residuals from regressions of log earnings (or log income) against observables, ξ ˆi = yi −g(ζ ˆ ;Xi ). In particular, for male earnings, we estimate least-squares regressions, a,t a,t t a,t separately for each year, of log earnings against a full set of age dummies, thus removing the predictable lifecycle earnings variation. For household income, we regress, separately for each year, log household income on a full set of age dummies for the primary tax (cid:28)ler, indicators of gender and marital status for the primary (cid:28)ler, and a full set of dummies for 17 the number of children (up to ten) in the household. Since the tax data do not contain information on race and education, the corresponding part of income variation will remain in the residuals and will add to the variation attributed to the permanent component. The Appendix describes our (cid:28)rst-stage regressions in more detail. In the second stage, we estimate all model parameters (other than ζ ) using a minimum t distance estimator that matches the model’s theoretical variances and autocovariances to their empirical counterparts. The error-components model in equations (2)-(5) implies a speci(cid:28)c parametric form for each variance and autocovariance of residual income, given normalized age a, calendar year t, and lead k. These theoretical variances and autocovariances, denotedbycov(a,t,k),arefunctionsofthemodelparametersσ2,σ2,σ2,σ ,σ ,σ ,σ2,ρ,θ,σ2, α β γ αβ αγ βγ r z and λ ,π for t = 1987,...,2006 . We estimate these model parameters by minimizing the t t distance between the theoretical variances and autocovariances implied by the model, and their empirical counterparts, which we compute from our longitudinal tax return data for a = 1,...,35, t = 1987,...,2006, and k = 0,...,19. This yields over 6,000 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. We do not use an optimal weighting matrix for reasons discussed in Altonji and Segal (1996). The basic intuition for identi(cid:28)cation of the permanent and transitory components is that the contribution of the transitory component to the autocovariance of income between two periods vanishes as the distance between the periods gets large enough. 17 In section 8 we examine the robustness of our results to alternative treatments of household size and composition. 15

4.2 Variance Decomposition After estimating our baseline model in equations (2)-(5), we can use it to determine its implications for annual cross-sectional income inequality for each income measure, whether at the individual or at the household level. In other words, we will use the estimated model to decompose the cross-sectional variance of log (residual) income. In particular, for each calendar year between 1987 and 2006, the model implies a speci(cid:28)c value for the total crosssectional variance, the permanent variance, and the transitory variance of log (residual) income, as a function of the model parameters and given an age distribution. We compute these variances implied by the estimated model, assuming an age distribution equal to the 18 actual empirical age distribution in our sample. Note that the evolution of the permanent variance and the transitory variance is primarily determined by the estimates of the λ and π t t parameters, respectively. We also repeat this procedure to derive inequality decompositions into permanent and transitory components for each one of the restricted models RM1, RM2, and RM3. 5 Male Earnings Inthissectionwepresentmodelestimatesformaleearnings,andweuseourestimatedmodels to decompose the cross-sectional variance of (residual) log male earnings into permanent and transitory components. We also extensively explore the sensitivity of our results to alternative model speci(cid:28)cations, and we provide a detailed discussion for the outcomes of the comparison across di(cid:27)erent models. 5.1 Baseline Model Estimates and Inequality Decomposition Table II presents estimates for our error-components models, baseline and restricted, for (residual) male earnings. Columns 1a and 1b display point estimates and standard errors for ourbaselinemodel. Wenotethatallmodelparametersarepreciselyestimated. Startingwith the permanent earnings component, the parameter estimates (other than the λ parameters) t are σˆ2 = .2487, σˆ2 = .0019, σˆ2 = .0000018, σˆ = −.0092, σˆ = .00024, σˆ = −.00006, α β γ αβ αγ βγ and σˆ2 = .0122. All of these parameter estimates are statistically signi(cid:28)cant, so the data r 18 We can also use the estimated model to compute similar decompositions for any age group, or for any age distribution. In fact, we have computed decompositions for several di(cid:27)erent age groups, but we do not showthoseresultshereduetospaceconsiderations. Focusingonalternativeagedistributionsleadstosimilar results. 16

appear to support the inclusion of both (quadratic) heterogeneous income pro(cid:28)les and a random walk in permanent earnings. For the transitory earnings component, the parameter estimates (other than the π parameters) are ρˆ = .6281, θ ˆ = −.3302, and σˆ2 = .1986. The t z estimate of ρ implies a relatively persistent transitory earnings process, despite the presence ofbothheterogeneousincomepro(cid:28)lesandarandomwalkinpermanentearnings. Inaddition, the data appear to support the presence of moving average innovations in the autoregressive ˆ transitory earnings (the estimate θ = −.3302 has a standard error of .0153). We will return to these points in the section 5.2. The inequality decomposition implied by our baseline model is presented in Figure II, panel (a). Here, the top line, which shows the total cross-sectional variance implied by the estimated model, is very close to the empirical cross-sectional variance of log (residual) male earnings in our sample. Hence, our baseline speci(cid:28)cation is su(cid:30)ciently (cid:29)exible to (cid:28)t the evolution of the cross-sectional variance over calendar time. It can also be seen that the baseline model attributes, on average, 65% of the total variance to the permanent component of earnings, and the remaining 35% to the transitory component. More importantly for our purposes, the permanent variance increases by 30% between 1987 and 2006, while the transitory variance (cid:29)uctuates over the twenty-year period, but does not increase, on net. In fact, the transitory variance is 2% lower in 2006, compared to 1987. In other words, the entire 17% increase in the total cross-sectional variance of (residual log) male earnings is driven by an increase in the permanent earnings variance, thus re(cid:29)ecting an increase in permanent inequality. 5.2 Robustness: Alternative Model Speci(cid:28)cations In Table II, Columns 2a through 4b show parameter estimates and standard errors for the restricted models RM1 (no heterogeneous pro(cid:28)les), RM2 (no random walk), and RM3 (no MA component). For each of the restricted models, all model parameters are precisely estimated. In Figure II, the corresponding inequality decompositions are presented in panels (b)-(d). As was the case for the baseline model, here too the total cross-sectional variance implied by each restricted model is very close to the empirical cross-sectional variance in our sample. This means that the restricted models are also (cid:29)exible enough to (cid:28)t the evolution of the cross-sectional variance over calendar time. Furthermore, the trends in the permanent and transitory variances are remarkably robust across model speci(cid:28)cations. In particular, all models (cid:28)nd that the permanent variance increases over the sample period, while the transitory variance does not, and also that the temporal pattern of the evolution of both 17

variances is remarkably similar. Thus, all speci(cid:28)cations imply that the entire increase in the total cross-sectional variance of log residual male earnings over the period 1987-2006 was driven by an increase in the permanent part of the variance, re(cid:29)ecting an increase in permanent inequality. However, the shares of the total cross-sectional variance attributed to the permanent and transitory components di(cid:27)er widely across the di(cid:27)erent models. Speci(cid:28)cally, the transitory share of the total variance, which was 35% for the baseline model, is, on average, 60% for RM1 (no heterogeneous pro(cid:28)les), 43% for RM2 (no random walk), and 27% for RM3 (no MA component). Given this range of results for the decomposition of inequality, we next proceed to examine the source of the di(cid:27)erences across model speci(cid:28)cations. We will argue that these di(cid:27)erences re(cid:29)ect di(cid:27)erences in the (relative) degree of persistence in transitory earnings across the various model speci(cid:28)cations. Intuitively, when transitory earnings are less persistent, then more of the persistence in the earnings data will be attributed to the permanent component, leading to a larger role assigned to permanent earnings overall. Conversely, when transitory earnings are more persistent, then more of the persistence in the data will be picked up by the transitory part, leading to a larger role played by the transitory earnings component. We begin by showing the di(cid:27)erences in the persistence of transitory earnings implied by the various estimated model speci(cid:28)cations. Table III shows, for each model, the fraction of a transitory shock that survives s periods after the shock. As column (1) shows, our ˆ baseline parameter estimates, ρˆ= .6281 and θ = −.3302, imply that 30%, 19%, and 5% of a transitory shock remains after 1, 2, and 5 years, respectively. That is, our baseline model implies a moderate degree of persistence in transitory earnings, despite the inclusion of both heterogeneous income pro(cid:28)les and a random walk component in permanent earnings. Restricted model RM1 (no heterogeneous pro(cid:28)les), by contrast, yields the estimates ρˆ= ˆ .9238 and θ = −.5912. The estimate of ρ, in particular, would appear surprisingly high for a transitory earnings component. Indeed, column (2) of Table III shows that these estimates imply that 33%, 31%, and 24% of a transitory shock remains 1, 2, and 5 years after the shock. Moreover, 16% of a transitory shock remains even 10 years after the shock. The reason for this high degree of persistence in transitory earnings is that permanent earnings in model RM1 lack a heterogeneous income pro(cid:28)les component, which would capture part of the persistence in the earnings data. Therefore, a larger share (relative to the baseline) of the persistence in the data is attributed here to the transitory component instead. At the opposite end, restricted model RM3 (no MA component) yields a very small 18

estimate of the autoregressive parameter, ρˆ = .2134, which implies very little persistence in transitory earnings. In fact, as shown by column (4) of Table III, only 5% percent of a transitory shock remains after 2 years, with essentially no e(cid:27)ect remaining 3 years after the shock. Note that, according to our baseline model, the e(cid:27)ect of transitory shocks falls ˆ rapidly after one period (via θ = −.3302), but decays more slowly after that (via ρˆ= .6281). By contrast, under model RM3’s restriction that θ = 0, the rapid fall of the e(cid:27)ects of a transitory shock after one period can only be captured by the autoregressive parameter ρ, which in turn implies that the estimate of ρ will be pushed downward. The above discussion illustrates a more general point that is often overlooked in discussions of permanent-transitory decompositions of income. In reality, incomes are subject to many di(cid:27)erent types of shocks. While some of these shocks might be truly permanent, and some truly transitory, many shocks are likely to have varying degrees of persistence, in between the two extremes. Decomposing income into permanent and transitory components requires taking a stand on what degree of persistence will be considered (cid:16)permanent(cid:17) and what degree will be considered (cid:16)transitory(cid:17). This choice necessarily involves some arbitrariness. Our approach here is to rely on a carefully speci(cid:28)ed error-components model that captures as well as possible the entire covariance structure of earnings, building on what has been learned about the dynamic properties of income from the literature on income dynamics. We thus advocate working with the model that best describes the data, which in our case is our baseline model. In order to further substantiate the claim that our baseline model best (cid:28)ts our data, we nextshowthattherestrictionsimpliedbymodelsRM1, RM2, andRM3areactuallystrongly rejected by the data. This implies that the restricted models miss important dimensions of the earnings data, despite capturing the evolution of total inequality and yielding results for inequality trends similar to the baseline model. In other words, it has to be the case that the di(cid:27)erent results across speci(cid:28)cations for the shares of the total cross-sectional variance attributed to the permanent and transitory components are due to aspects of the restricted models that are not supported by the data. Starting with model RM1 (no heterogeneous pro(cid:28)les),ajointtestoftherestrictionsimposedbythemodel,namelyσ2 = σ2 = σ = σ = β γ αβ αγ σ = 0, yields the F statistic F = 71.15, which overwhelmingly rejects those restrictions βγ (the critical F value at a 1% level is 3.02). Similarly, a test of the restriction σ2 = 0 imposed r by model RM2 (no random walk) yields a t statistic of t = 2.17, which rejects this restriction at a 5% level (though not at a 1% level). Finally, a test of the restriction θ = 0 imposed by RM3 (no MA component) yields the t statistic t = −7.53, rejecting the null at a 1% level. 19

Overall, our results support the inclusion of a (quadratic) heterogeneous income pro(cid:28)les component, of a random walk component, and of moving-average innovations in the (autoregressive) transitory component in models of individual male labor earnings. The presence of quadratic heterogeneous income pro(cid:28)les is especially strongly supported in our data. As we show in Figure A.1 of the Appendix, model RM1’s restriction of no heterogeneous income pro(cid:28)les leads to a poorer (cid:28)t in the evolution of the cross-sectional variance of earnings over the lifecycle. In addition, this restriction leads to the largest di(cid:27)erence, relative to the baseline model, in the permanent and transitory shares of the cross-sectional variance in Figures II (a)-II (d). Although the data also support the inclusion of a random walk component in the error-components model, that support is somewhat weaker, as evidenced by the (cid:28)nding above that model RM2’s restriction of no random walk is rejected at a 5% level, but not at a 1% level. Furthermore, section 7 shows that, when working with total household income, we cannot reject model RM2’s restriction of no random walk. 6 Comparison to Alternative Decomposition Methods In this section, we expand our analysis to explore simpler, approximate decomposition methods that do not rely on models and that have been used in the literature to examine the permanent-vs-transitory nature of inequality. We also investigate the relation across the di(cid:27)erent inequality-decomposition methods, model-based as well as non-model-based, and we propose an explanation for the di(cid:27)erences across methods. By clarifying the connections across methods, we hope to propose one way of thinking about the variety of results they yield as well as some guidance for the potential outcomes of di(cid:27)erent methodological choices. 6.1 KSS and BPEA methods Here, we consider two approximate inequality-decomposition methods which basically de(cid:28)ne permanent income as the average of annual income over a certain period of time, and then transitory income as the deviations of annual income from that average. The (cid:28)rst is a simple and intuitive method that does not explicitly rely on any model. This method, used in Kopczuk, Saez, and Song (2010) and referred to as ‘KSS’ here for convenience, de(cid:28)nes person i’s permanent earnings in year t as the average of person i’s annual log earnings (or residual log earnings) over a P-year period centered around t. Transitory earnings for person i in year t are then de(cid:28)ned as the di(cid:27)erence between person i’s current annual earnings at t and permanent earnings in the same year. The permanent and transitory variances are 20

next calculated as the variances, across individuals, of permanent and transitory earnings, respectively. Figure III (a) shows the decomposition of the cross-sectional variance of (residual) male earningsintopermanentandtransitorypartsusingtheKSSmethodwithparameterP = 5. 19 Two points are worth noting here. First, in terms of the trends, the increase in the crosssectional variance is again entirely driven by the permanent component, as it was in our model-based results. Second, in terms of the relative shares, this method attributes on average 87% of the cross-sectional variance to the permanent component, and only 13% to the transitory component, as opposed to 65% and 35%, respectively, in our baseline model. In other words, the KSS method attributes an overwhelmingly large part of the variance to the permanent earnings component. Inordertoseewhythisisthecase, notethattheKSSdecompositiondependscruciallyon the value of P(cid:22)the number of years used to de(cid:28)ne permanent earnings. We show in Figure A.2 of the Appendix that, as P increases, the permanent share falls and the transitory share rises. For example, for P = 3,5,7 and 9 years, the transitory share is 8%, 13%, 16%, and 18%, respectively. The choice of P is obviously arbitrary. Nonetheless, using large values of P is impractical, as the construction of the decomposition leads to the loss of data at the endpoints of the sample (for example, using P = 11 would lead to the loss of 5 years 20 of data at each endpoint of our 20-year sample). Furthermore, in the KSS decomposition, (cid:16)transitoryearnings(cid:17) captureonlypurelytransitoryearnings(withnopersistence). However, as we have shown, the data provide evidence of moderate persistence in transitory earnings. We also consider a second approximate variance decomposition method, which was introduced by Gottschalk and Mo(cid:30)tt (1994). Following Mo(cid:30)tt and Gottschalk (2008), we refer to this method as ‘BPEA’. The BPEA method is similar, though not identical, to the KSS, and we consider it separately because it relies on a simple model of income, which provides a 21 more direct way of relating it to our error-components models. Speci(cid:28)cally, BPEA is based 19 Kopczuk, Saez, and Song (2010) use P = 5. They use raw (as opposed to residual) log earnings and restrict observations to individuals who are present in the sample for all (cid:28)ve years. We use residual log earningsanddonotrequireindividualstobepresentinall(cid:28)veyears. However,theresultsarenotmaterially di(cid:27)erent when we follow their treatment and restrictions. 20 Atthelimit, onewouldde(cid:28)nepermanentearningsasaverageearningsoveraperson’sentirecareer(say, 35 years), and transitory earnings as the deviation of current earnings from average career earnings. Under this de(cid:28)nition, however, it makes little sense to talk about changes over time in the relative importance of a person’s permanent income, and it would not be possible to construct a series of such decompositions over time with the available data. 21 The di(cid:27)erence between the BPEA and the KSS methods essentially re(cid:29)ects a (cid:16)bias correction term(cid:17)in the random e(cid:27)ects formula upon which the BPEA decomposition is based. For details, see Gottschalk and Mo(cid:30)tt (2009). 21

on the simple speci(cid:28)cation of (residual) log earnings ξ = α +ε , where α is purely permait i it i nent (time-invariant) and ε is purely transitory (iid). For a P-year window centered around it each year t, BPEA uses the standard formulas from this simple (cid:16)random e(cid:27)ects model(cid:17) to compute the permanent variance of ξ as the variance of α , and the transitory variance of it i ξ as the variance of ε . To obtain a series of permanent and transitory variance estimates it it over time, this procedure is repeated for consecutive, overlapping P-year moving windows. Figure III (b) presents the BPEA inequality decomposition. Once again, the decomposition implies that the entire increase in the cross-sectional variance is driven by an increase in the permanent variance. This method (with P = 5) attributes about 80% of the total crosssectional variance to the permanent component, slightly less than the KSS decomposition, but still quite a bit more than our baseline error-components model. Again, the reason for this di(cid:27)erence lies in the simple structure of the model underlying the BPEA decomposition. In particular, note that our baseline model of equations (2)-(5) essentially nests the simpler model upon which BPEA is based, with restrictions σ2 = σ2 = σ = σ = σ = σ2 = 0 β γ αβ αγ βγ r in the permanent component, and ρ = θ = 0 in the transitory component (and with our baseline model using the more (cid:29)exible λ and π for t = 1987,...,2006, rather than the Pt t year moving windows). As our results in section 5.2 indicated, these restrictions are strongly 22 rejected in the data. The discussion above suggests that the reasons for the inequality-decomposition di(cid:27)erences between our baseline model and the non-model-based methods considered here are in fact closely related to the reasons for the di(cid:27)erences across di(cid:27)erent model speci(cid:28)cations. Speci(cid:28)cally, the KSS and BPEA approximate methods do not allow for persistence in transitory income. As a result, they attribute to the permanent income component part of what is in reality a transitory (though serially correlated) shock, thereby overstating the importance of the permanent part of inequality. In other words, the simpler decompositions rely on restrictions that are strongly rejected by the data. Therefore, our analysis here indicates that the search for the appropriate inequality-decomposition method needs to carefully consider the nature of the data, with particular emphasis on the relative degree of persistence in the transitory component of income. 22 Mo(cid:30)tt and Gottschalk (2008) also favor the use of richer error-components models over the simple BPEA decomposition. 22

6.2 Volatility Next, we examine the evolution of the standard deviation of changes in male earnings over short horizons. Following Shin and Solon (2011), we refer to this measure as earnings volatility. Thismeasureofdispersioninthecross-sectionaldistributionofshort-runincomechanges 23 isrelated, thoughnotequivalentto, theconceptofthetransitoryvariance. FigureIVshows the evolution of the standard deviation of one-year (the lower line) and two-year (the upper line) percent changes in (residual) male earnings. As the (cid:28)gure shows, we (cid:28)nd no clear increasing or decreasing trend in male earnings volatility over our sample period. This is consistent with the stable transitory variance of male earnings found in the rest of our decompositions. This (cid:28)nding thus reinforces the result that the increase in male earnings inequality over 1987-2006 was of a permanent nature, as the transitory variance and the volatility of male earnings appear to be overall (cid:29)at during this period. 7 Household Income In this section we examine the evolution of the permanent and transitory variance of total household income. As already mentioned in section 3, we carry out the analysis using two alternative samples. First, our ‘male earnings’ sample, which is identical to the sample used in our analysis of male earnings. This sample consists of households with a male primary (cid:28)ler aged 25-60, whose annual labor earnings are above the minimum threshold. Second, our ‘full’ household sample, which mostly adds households with a female primary (cid:28)ler (typically 24 single females). As Table I shows, the full sample has 105,544 observations more than the male earnings sample. The analysis here is performed on residuals from a (cid:28)rst-stage regression of log household income on gender, age and (cid:28)ling status of the primary (cid:28)ler, and on a full set of dummies for the number of children (see the Appendix for details). In section 8 we investigate the robustness of our results to alternative treatments of household size and composition. 23 In particular, for most speci(cid:28)cations of an earnings process, volatility and the transitory variance will tend to move together. See the discussion in Shin and Solon (2011) for details on the relation between volatility and transitory variance. 24 It also adds some households for which labor earnings of the male primary (cid:28)ler are below the minimum threshold, but for which total household income is above the minimum threshold. 23

7.1 Pre-Tax Household Income Table IV presents point estimates and standard errors for our error-components models estimated on total pre-tax household income data, for both our ‘male earnings’ sample and our broader ‘full’ household sample. Columns 1a-1b show estimates of our baseline model on the male earnings sample. Note that the estimate of the random walk innovation variance σˆ2 = .0014 (.0059) is not statistically signi(cid:28)cant. That is, when using our male r earnings sample, we cannot reject the hypothesis that σ2 is in fact zero. For this reason, we r will also present results for restricted model RM2, which imposes σ2 = 0. 25 We have also r estimated restricted models RM1 and RM3, but do not show the results here because of space constraints. Columns 2a-2b of Table IV present point estimates and standard errors for model RM2 estimated on our male earnings sample. Note that for this speci(cid:28)cation, all parameter estimates are statistically di(cid:27)erent from zero. Figures V (a) and V (b) present the variance decompositions for our baseline model 26 and for model RM2, both estimated on our male earnings sample. Of course, since we cannot reject the hypothesis that σ2 = 0, the decompositions for the two models are almost r identical. In particular, the transitory variance accounts for about 40% of the total variance, which is similar to our (cid:28)nding for male earnings. Moving to the time trends, the transitory variance increased by about 30% over the sample period, mainly in the early 2000s. The permanent variance rose overall by about 45%, showing (cid:28)rst a relatively steady increase until around 2000, followed by a moderate decline in the early 2000s and then a resumed increase in the last three years of the sample. All told, the transitory variance contributed about one third of the increase in the total cross-sectional variance. Thus, as in the case of male labor earnings, most of the increase in household income inequality (two thirds) represented an increase in permanent inequality. However, in contrast to male earnings, the transitory variance here did play a role in the increase in household income inequality. FigureVIshowstheevolutionofthestandarddeviationofone-year(thebottomline)and two-year (the top line) percentage changes in total household income on our male earnings 25 Whenusingthefullhouseholdsample,werejectσ2 =0. However,ingeneral,thehouseholdincomedata r provide less support than male labor earnings for the inclusion of a random walk component in permanent income. In particular, whether or not the restriction σ2 =0 can be rejected depends on factors such as the r speci(cid:28)c sample used, the level of the minimum threshold, whether the income data are before or after taxes, and so on. 26 Note that the reason why the total variance of household income in Figure V is lower in any given year than the total variance of male earnings shown earlier 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 the variance of male earnings, as seen in Figure I. 24

sample. As shown by the (cid:28)gure, household income volatility rose 13% for one-year income changes and 12% for two-year income changes, over the sample period. This provides further evidence that the transitory variance did in fact contribute to the increase in the crosssectional inequality in the case of household income. Furthermore, we have also computed (but do not show) variance decompositions for restricted models RM1 (no heterogeneous pro(cid:28)les) and RM3 (no MA component), as well as for the KSS and BPEA methods, all of which con(cid:28)rm this result. In going from individual male earnings to total household income, a number of income components are added. We group these components into four main categories: spousal labor earnings, transfer income, investment income, and business income. Transfers are de(cid:28)ned here as the sum of alimony received, pensions and annuities, unemployment compensation, social security bene(cid:28)ts, and tax refunds. Investment income includes interest, dividends and capital gains. Business income includes income from self-employment, from partnerships, 27 and from S-corporations. Next, we examine which component or category of household income is responsible for the increase in the transitory variance of total household income. We start with male labor earnings, and then sequentially (and cumulatively) add each of the other categories of income, namely spousal labor earnings, transfer income, investment income, and business income. Foreachoftheresultingincomeaggregates, weestimateourerror-componentsmodels, 28 and we decompose the cross-sectional variance into permanent and transitory parts. The decompositions, based on restricted model RM2 (no random walk) and our male earnings sample, are presented in Figure A.3 of the Appendix. The other restricted models lead to similar conclusions. Starting with male earnings, and moving along the series of increasingly broad income aggregates, the changes in the transitory variance between 1987 and 2006 are -2%, 8%, 17%, 29%, and 28%, respectively. This takes us from the slight decrease in the transitory variance of male earnings, to the 28% increase in the transitory variance of total household income that we found earlier. We conclude that each of spousal labor earnings, transferincome, andinvestmentincomecontributedtotheincreaseinthetransitoryvariance of total household income. Moreover, none of these categories appears to have played a par- 27 Using the full household income sample, and on average over 1987-2006, male labor earnings account for about 50% of total household income, female labor earnings for 26%, retirement and transfer income for 5%, investment income for 7%, and business income for 12%. 28 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 di(cid:30)cult to estimate error-components models separately for each income category. 25

ticularly dominant role in driving the increase in the transitory variance of total household 29 income. Columns 3a-4b of Table IV present point estimates and standard errors for the baseline and restricted model RM2 using our broader ‘full’ household income sample. The corresponding variance decompositions are presented in Figures VII (a) and VII (b). The results are similar to those obtained when using the male earnings sample. In particular, the transitory variance increased over 1987-2006, contributing about one third of the increase in the total cross-sectional variance. Results (not shown) again indicate that various sources of household income contributed to the increase in the transitory variance, with no single source playing a particularly prominent role. 7.2 The Role of the Federal Tax System This section explores the role of the federal tax system in the evolution of income inequality. In particular, we examine whether the evolution of inequality for after-tax household income di(cid:27)ers materially from the evolution of inequality for pre-tax income. As discussed in section 3.1, our measure of after-tax household income re(cid:29)ects all federal personal income taxes, including all refundable tax credits such as the earned income tax credit and the child tax credit, as well as payroll taxes. Figure VIII shows the evolution of the total, permanent, and transitory variance of pretax household income (the solid lines), along with the corresponding variances of after-tax household income (the dashed lines), based on our male earnings sample. As the (cid:28)gure shows, the variance of after-tax income is on average 15% smaller than the variance of pretax income, re(cid:29)ecting the overall progressivity of the U.S. federal tax system. The e(cid:27)ect of the tax system in reducing income inequality appears fairly stable over the sample period, although it might have increased marginally around 1996. Overall, however, the tax system does not seem to have signi(cid:28)cantly altered the trend toward rising inequality: The variance of (residual) pre-tax income in Figure VIII increased by 37% over our sample period, while the variance of (residual) after-tax income increased by 35%. We reach similar conclusions when we use our full household sample. The (cid:28)nding of little change in the e(cid:27)ect of the federal tax system on the evolution of inequality in recent years might appear surprising in light of the well publicized reductions in marginal tax rates, especially at the high end of the income distribution, in 2001 and 2003. 29 Investment income here includes capital gains. However, we have veri(cid:28)ed that excluding capital gains leads to similar conclusions, in that the transitory variance of investment income contributes to the increase in the transitory variance of total household income even if capital gains are excluded. 26

However, such changes in top marginal tax rates were accompanied by (smaller) reductions in marginal tax rates for other income groups as well as by signi(cid:28)cant expansions of the earned income tax credit and the child tax credit. Our results suggest that the net e(cid:27)ect of all changes to the federal tax system was small for purposes of the evolution of income inequality. 8 Robustness Tests 8.1 Changes Over Time in the Age Distribution Our error-components models imply that the decomposition of the cross-sectional variance of income into permanent and transitory components depends on age. This, in turn, means that the permanent-transitory variance decomposition in a given calendar year will depend on the age distribution in that year. Therefore, one possible concern is that changes over time in the age distribution might a(cid:27)ect the decomposition, masking the e(cid:27)ects of ‘true’ 30 changes in the variance of permanent and transitory income components. To address this issue, we reweigh the moments matched in our estimation procedure so as to keep the age distribution constant over time. Our methodology is an extension of the one introduced by DiNardo, Fortin, and Lemieux (1996), and it is described in detail in the Appendix. Overall, our results under this reweighing procedure are essentially unchanged, both for male earnings (see Figure A.4 (a) of the Appendix) and for total household income. We conclude that our model-based (cid:28)ndings are not materially a(cid:27)ected by changes over time in the age distribution of the taxpayer population. 8.2 Changes Over Time in the Distribution of Household Composition In the case of household income, an additional concern is that our results might be a(cid:27)ected by changes in the distribution of household composition over time. For instance, if total income is more variable for married households than for single households, then changes in the married-vs-single composition of the taxpayer population over time could a(cid:27)ect trends in the variances. Although our baseline household-income treatment does control for the e(cid:27)ects of changes in household composition on the mean of household income, it does not 30 Table A.1 of the Appendix shows the mean and standard deviation of the age distribution in each calendar year for both our male earnings sample and our full household sample. 27

control for potential e(cid:27)ects on the variance. In order to check that our results for household income are not just capturing changes in the distribution of household composition, we have performed the following three tests. First, following the approach described in section 8.1, we reweigh the moments matched in estimation in such a way as to keep the distribution of household composition unchanged (see Figure A.4 (b) of the Appendix). Second, we restrict the household income sample to married households only. Third, we treat observations as coming from di(cid:27)erent households 31 whenever a household (couple) forms or splits. In all three tests described above, our main results remain essentially unchanged. In particular, we continue to (cid:28)nd that the increase in male earnings inequality over our sample period was entirely driven by an increase in permanent inequality, while the increase in household income inequality was predominantly permanent, though partly also re(cid:29)ecting an increase in transitory inequality. 8.3 Changes in the Minimum Threshold We have also examined the sensitivity of our results to alternative minimum thresholds for income. Recall that our analysis thus far excluded person-year observations where annual earnings or household income were below one-fourth of a full-year, full-time minimum wage. We have experimented with both lower (up to one-half of the original threshold) and higher (up to two times the original threshold) minimum thresholds. In all cases, our main results are mostly unchanged. In particular, the increase in male earnings inequality is still entirely driven by an increase in permanent inequality, while the increase in household income inequality is predominantly, but not entirely, permanent. The shares of the total variance attributed to the permanent and transitory components, however, are somewhat sensitive to setting the minimum threshold to a larger value than in our main treatment (see Figures A.5 (a) and A.5 (b) in the Appendix). 31 That is, we de(cid:28)ne a new sample in which households with di(cid:27)erent size/composition are treated as separate households. For example, if person A is observed for (cid:28)ve years, then person A marries person B andthecoupleisobservedfor(cid:28)veyears,andthenthecouplesplitsandpersonAisobservedforanother(cid:28)ve years,wetreatthesethreedi(cid:27)erent(cid:28)ve-yearspellsforpersonAasobservationsonthreedi(cid:27)erenthouseholds. Sinceweareconcernedwithhouseholdincome(asopposedto,say,consumption),wefocusontheformation and dissolution of couples, and abstract from changes in household size and composition having to do with children. 28

9 Conclusions We use a large panel of household income to analyze the role of permanent and transitory income components in the evolution of inequality in male labor earnings and total household income in the United States over the period 1987-2006. We (cid:28)rst document an increase in inequality in male earnings and pre-tax and after-tax household income in our tax return dataset during this period, consistent with what other studies have documented on di(cid:27)erent datasets. We then examine the role of permanent and transitory income components for the increase in inequality, as measured by the cross-sectional variance of log income. The quality and signi(cid:28)cant size of our dataset allow us to start the analysis by precisely estimating rich non-stationary error-components models of income dynamics. One of the main advantages of error-components models is that they are su(cid:30)ciently detailed and (cid:29)exible to be able to capture many facets both of the autocorrelation of earnings and of the evolutionofearningsoverthelifecycle. Indeed, ourmainspeci(cid:28)cationallowsforapermanent income component with quadratic heterogeneous income pro(cid:28)les and a random walk process, with the relative importance of the permanent component allowed to vary over calendar time. The transitory income process is speci(cid:28)ed as an ARMA(1,1) process with year-speci(cid:28)c innovations variances. We also expand our analysis to explore simpler, approximate inequality decomposition methods, and we propose an explanation for the connections between the model- and non-model based methods. Overall, we (cid:28)nd remarkably robust results for the trends of the permanent and transitory variancecomponents. Formalelaborearnings,we(cid:28)ndthatthepermanentvarianceincreased over the sample period, while the transitory variance did not. Hence, the increase in male earnings inequality was driven entirely by the permanent component, thus re(cid:29)ecting an increase in permanent inequality. For household income, both before and after taxes, the increase in inequality over this period was predominantly, but not entirely, permanent, with the transitory component contributing about one third of the increase in inequality. This increase in the transitory variance of total household income re(cid:29)ects an increase in the transitory variance of components such as spousal earnings, transfer income, and investment income. We also (cid:28)nd evidence that the U.S. federal tax system played an important role in reducing the level of income inequality over our sample period, but it did not signi(cid:28)cantly alter the broad trends toward increasing inequality. In contrast to the trends, we show that the shares of the total cross-sectional variance attributedtothepermanentandtransitoryincomecomponentsaresensitivetothedecomposition method used and, in the case of model-based decompositions, to model speci(cid:28)cation. 29

We provide evidence indicating that one of the main reasons for these di(cid:27)erences across methods pertains to the degree of relative persistence they allow for in the transitory earningsprocess. Intuitively, whentransitoryearningsaremore(less)persistent, thenless(more) of the persistence in the data will be attributed to the permanent component, leading to a smaller (larger) role assigned to permanent earnings overall. Since the simpler methods do not allow for persistence in transitory income, they attribute to the permanent income component part of what is in reality a transitory (though serially correlated) shock, thereby overstating the importance of the permanent part of inequality. In other words, the simpler decompositions rely on restrictions that are strongly rejected by the data, and hence produce erroneous inequality decompositions when the true underlying data generating process is rich. Therefore, our analysis provides signi(cid:28)cant guidance for researchers deciding between alternative permanent-transitory decomposition methods, and it suggests that the search for the appropriate method needs to carefully consider the nature of the data, with particular emphasis on the relative degree of persistence in the transitory component of income. 30

Appendix Creating residuals from (cid:28)rst-stage regression We focus on residual earnings (or income) variation, namely on the part of the earnings (income) variance that is not explained by observable characteristics of the individual or household. We construct residual individual labor earnings by applying least squares (separately for each year) to a regression of log earnings against a full set of age dummies. This regression purges individual earnings from the predictable lifecycle variation that is common to all individuals, and from the e(cid:27)ect (on the mean) of economy-wide factors (‘year e(cid:27)ects’). The regression for individual earnings, yi , is thus: a,t yi = f(c1,Ai ) , a,t t a,t where c1 is a year-speci(cid:28)c constant and Ai is a full set of age dummies. t a,t Similarly, we construct residual household income by applying least squares (separately foreachyear)toaregressionofloghouseholdincomeagainstafullsetofagedummiesforthe primary (cid:28)ler, gender of the primary (cid:28)ler, and indicators for household size and composition. The latter include an indicator of whether the primary (cid:28)ler is married or single, and a full set of dummies for the number of children (up to ten) in the household. The regression for household income, yh , is thus: a,t yh = g(c2,Mh,Ah ,Fh ) , a,t t a,t a,t wherec2 isayear-speci(cid:28)cconstant, Mh isadummyformale, Ah isafullsetofagedummies, t a,t and Fh is a full set of family size/composition dummies. a,t Moment conditions Letabe"normalizedage"or"potentialexperience", de(cid:28)nedasa = age−25, oryearselapsed sinceage25. Then, the theoreticalmomentsimpliedbyourbaselineerror-componentsmodel in equations (2)-(5) are as follows: 31

cov(ξi ,ξi ) = λ ·λ ·(σ2 +σ2 ·a·(a+k)+σ2 ·a2 ·(a+k)2 a,t a+k,t+k t t+k α β γ +σ ·(2a+k)+σ ·(2a2 +2ak +k2) αβ αγ +σ ·(2a3 +3a2k +ak2)+a·σ2) βγ r +ρkvar(z ) a,t +1[k ≥ 1]·ρk−1 ·θ·π2 ·σ2 . t z For t = 1987, 2 ≤ a ≤ 35: 1−ρ2(a−1) var(z ) = π2 σ2 +(ρ+θ)2σ2 . a,1987 1987 z z 1−ρ2 For 1987 ≤ t ≤ 2006, a = 1: var(z ) = π2 ·σ2 . 1,t t z For 1988 ≤ t ≤ 2006, 2 ≤ a ≤ 35: var(z ) = ρ2var(z )+σ2 ·(π2 +θ2 ·π2 +2·ρ·θ·π2 ) . a,t a−1,t−1 z t t−1 t−1 To obtain identi(cid:28)cation, we impose the normalization λ = π = 1 for all calendar t t years t ≤ 1987, where 1987 is the (cid:28)rst year in the sample. Additionally, we impose the normalization π = π , since λ and π cannot be identi(cid:28)ed separately in the last year 2005 2006 t t of the sample, t = 2006. KSS and BPEA methods In the KSS methodology, the permanent variance in year t is var(1 (cid:80)t+k ξ ), where ξ is P j=t−k ij it the relevant measure of (log) earnings and k = P−1, and the transitory variance is var(ξ − 2 it 1 (cid:80)t+k ξ ). Following Kopczuk, Saez, and Song (2010), we set P = 5. P j=t−k ij In the BPEA methodology, let ξ be residual log earnings, N the number of individuals, it ¯ T ≤ P the number of years (within the P-year window) that person i is observed, ξ the i i ˜ person-speci(cid:28)c average earnings over T years, ξ the mean of log earnings across the full sami ¯ ple, and T the mean years covered by the window over the individuals in the sample. Then, (cid:104) (cid:105) the exact formulas (within each (cid:28)xed-size window) are σˆ2 = 1 (cid:80)N 1 (cid:80)Ti (ξ −ξ ¯ )2 ε N i=1 Ti−1 t=1 it i for the transitory variance, and σˆ2 = 1 (cid:80)N (ξ ¯ −ξ ˜ )2 − σˆ ε 2 for the permanent variance. α N−1 i=1 i T¯ 32

The permanent and transitory variance components from BPEA are very similar, though not identical, to the KSS ones. The main di(cid:27)erence lies in the presence of the term −σ ε 2 in T¯ the permanent BPEA variance. See Gottschalk and Mo(cid:30)tt (2009), footnote 2. Note that Gottschalk and Mo(cid:30)tt use P = 9 (compared to our P = 5 in the main text). Thisslightlyincreasestheshareofthetotalvarianceattributedtothepermanentcomponent, and therefore slightly reduces the share attributed to the transitory component, but has no e(cid:27)ect on the trends of the two components. Reweighing the moments in estimation Thissectiondescribestheprocedurebywhichwereweighthemomentsusedinourestimation so as to keep the distribution of age and of household composition constant over time. This is an extension of the methodologies used in DiNardo, Fortin, and Lemieux (1996), Lemieux (2006), and Altonji, Bharadwaj, and Lange (2010). It involves calculating weights such that the sample characteristics, when the sample is reweighed, are similar to those in a set of base years. We choose 1999 through 2001 to be the base years to which we wish to reweigh each of the individual years. The method proceeds as follows. We (cid:28)rst estimate a logit equation, where the dependent variable is an indicator variable for the observation coming from one of the base years, and the independent variables are a full set of age dummies (for household income, we also include indicator variables for being a single male, a single female, and for the number of children, uptoten). Wethenestimatetwentyseparatelogits, oneforeachyearofthesample, where the dependent variable is an indicator for the observation coming from that year, and the independent variables are the same as in the (cid:28)rst logit. Using the results from these logits, we then calculate the predicted probability that the observation came from one of the base years, given the demographic characteristics of the observation (denoted p(base years | z)), and the predicted probability that the observation came from the year that it actually came from, given demographics (p(year=t | z)). Given the unconditional probabilities in the sample that an observation came from a base year (p(base years)) or from a particular year (p(year=t)), the weight for an observation from year t is calculated as: p(base years | z)·p(year = t) Ψ(z) = . p(year = t | z)·p(base years) 33

Lifecycle Pro(cid:28)le of Earnings Variance This section illustrates the role of the (quadratic) heterogeneous income pro(cid:28)les component in (cid:28)tting the lifecyle pro(cid:28)le of the variance of earnings. Figure A.1 of the Appendix shows the evolution of the cross-sectional variance of male labor earnings over the lifecyle in our data, as well as the evolution implied by our estimated baseline model and our estimated restricted model 1 (no heterogeneous pro(cid:28)les). To construct the series labelled (cid:16)data(cid:17), we computed the variance of male labor earnings for each combination of normalized age a and calendar year t, and regressed it against a full set of year and age indicators. The (cid:16)data(cid:17) series displays the estimated coe(cid:30)cients on the normalized age indicators. As the (cid:28)gure shows, the lifecycle variance pro(cid:28)le is linear to concave in the early part of the lifecycle, and convex in the later years. The variance pro(cid:28)le constructed controlling for cohort e(cid:27)ects (not shown), rather than year e(cid:27)ects, is similarly shaped. The dotted and dashed lines in the (cid:28)gure display the evolution of the earnings variance over the lifecycle implied by our baseline model and restricted model 1 (no heterogeneous pro(cid:28)les). Note that the baseline model (cid:28)ts the lifecycle variance pro(cid:28)le very well, while restricted model 1 misses the variance in the (cid:28)rst few years and in the last few years of the lifecycle. Restricted models 2 (no random walk) and 3 (no MA component) imply a lifecycle variance pro(cid:28)le very similar to that of our baseline model, and are therefore not shown in the (cid:28)gure. 34

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Table I Descriptive Statistics by Calendar Year ‐ Various Income Measures Year Male Earnings Pre‐Tax Household Income After‐Tax Household Income Full Household Income Full Household Income Male Earnings Sample Male Earnings Sample Sample Sample Obs. Mean St Dev Obs. Mean St Dev Obs. Mean St Dev Obs. Mean St Dev Obs. Mean St Dev 1987 8,177 10.38 0.78 8,177 10.66 0.77 12,767 10.46 0.84 8,177 10.48 0.73 12,764 10.30 0.78 1988 8,681 10.35 0.81 8,681 10.66 0.80 12,991 10.47 0.86 8,681 10.48 0.76 12,978 10.30 0.80 1989 9,021 10.33 0.81 9,021 10.64 0.82 13,242 10.46 0.86 9,021 10.46 0.77 13,227 10.29 0.81 1990 9,092 10.33 0.81 9,092 10.63 0.81 13,353 10.45 0.86 9,092 10.45 0.77 13,341 10.29 0.81 1991 8,905 10.31 0.81 8,905 10.61 0.82 13,395 10.43 0.87 8,905 10.43 0.77 13,377 10.27 0.81 1992 8,923 10.32 0.83 8,923 10.62 0.84 13,480 10.44 0.88 8,923 10.45 0.79 13,464 10.28 0.83 1993 9,273 10.29 0.84 9,273 10.61 0.84 13,654 10.43 0.89 9,273 10.45 0.79 13,650 10.27 0.83 1994 9,387 10.30 0.83 9,387 10.63 0.84 13,838 10.43 0.89 9,387 10.45 0.80 13,821 10.26 0.84 1995 9,575 10.31 0.83 9,575 10.64 0.85 14,148 10.44 0.91 9,575 10.46 0.80 14,125 10.27 0.85 1996 9,624 10.33 0.83 9,624 10.66 0.86 14,257 10.45 0.92 9,624 10.48 0.81 14,233 10.29 0.86 1997 9,534 10.35 0.82 9,534 10.67 0.87 15,150 10.42 0.93 9,534 10.50 0.80 15,149 10.28 0.84 1998 9,762 10.38 0.82 9,762 10.70 0.88 15,515 10.46 0.94 9,762 10.54 0.82 15,525 10.32 0.86 1999 9,877 10.41 0.82 9,877 10.74 0.88 15,721 10.49 0.95 9,877 10.58 0.82 15,730 10.35 0.86 2000 9,904 10.43 0.82 9,904 10.76 0.88 15,918 10.51 0.95 9,904 10.59 0.82 15,923 10.37 0.87 2001 9,950 10.44 0.82 9,950 10.76 0.87 16,114 10.50 0.94 9,950 10.60 0.81 16,119 10.37 0.85 2002 9,860 10.43 0.84 9,860 10.76 0.88 16,095 10.50 0.94 9,860 10.61 0.81 16,104 10.38 0.85 2003 9,802 10.41 0.84 9,802 10.74 0.88 16,111 10.48 0.94 9,802 10.60 0.82 16,121 10.37 0.86 2004 9,956 10.42 0.84 9,956 10.74 0.89 16,296 10.49 0.96 9,956 10.61 0.83 16,310 10.38 0.88 2005 9,998 10.41 0.84 9,998 10.74 0.90 16,397 10.49 0.96 9,998 10.60 0.84 16,403 10.38 0.88 2006 10,123 10.42 0.85 10,123 10.76 0.91 16,526 10.51 0.96 10,123 10.62 0.85 16,546 10.40 0.89 Total (or Average) 189,424 10.37 0.83 189,424 10.69 0.86 294,968 10.46 0.91 189,424 10.52 0.80 294,910 10.32 0.84 Note: The slightly different number of observations of household income before and after taxes (in Full Household Income Sample) is due to the minimum income threshold in our sample selection criteria. This threshold is applied (separately) to both before‐ and after‐tax income.

Table II Estimates of Error‐Components Models, Male Earnings Column 1a 1b 2a 2b 3a 3b 4a 4b Baseline Model Restricted Model RM1 Restricted Model RM2 Restricted Model RM3 Parameter Estimate S.E. Estimate S.E. Estimate S.E. Estimate S.E. Permanent Component σ2 α 0.2487 0.0124 0.1337 0.0069 0.2250 0.0074 0.2813 0.0075 σ2 β (x100) 0.1902 0.0359 0.2599 0.0138 0.1082 0.0178 σ2 γ (x10,000) 0.0180 0.0014 0.0198 0.0012 0.0167 0.0012 σ αβ (x100) ‐0.9179 0.1975 ‐0.5954 0.0916 ‐1.5099 0.1083 σ αγ (x1,000) 0.2395 0.0844 0.1133 0.0418 0.4971 0.0485 σ βγ (x10,000) ‐0.5980 0.0512 ‐0.6720 0.0406 ‐0.5450 0.0410 σ2 r 0.0122 0.0056 0.0028 0.0003 0.0277 0.0018 λ 87 1.0000 1.0000 1.0000 1.0000 λ 88 1.0283 0.0133 1.0800 0.0312 1.0297 0.0145 1.0288 0.0122 λ 89 1.0556 0.0140 1.1474 0.0328 1.0610 0.0153 1.0537 0.0129 λ 90 1.0440 0.0134 1.1222 0.0322 1.0478 0.0148 1.0445 0.0122 λ 91 1.0585 0.0140 1.1523 0.0351 1.0651 0.0158 1.0557 0.0125 λ 92 1.0792 0.0148 1.1899 0.0373 1.0883 0.0169 1.0742 0.0131 λ 93 1.0581 0.0136 1.1483 0.0347 1.0659 0.0157 1.0550 0.0118 λ 94 1.0535 0.0137 1.1449 0.0353 1.0614 0.0159 1.0505 0.0118 λ 95 1.0695 0.0139 1.1868 0.0367 1.0798 0.0163 1.0653 0.0120 λ 96 1.0774 0.0137 1.2061 0.0372 1.0901 0.0162 1.0709 0.0118 λ 97 1.0743 0.0139 1.2094 0.0379 1.0884 0.0163 1.0660 0.0119 λ 98 1.0740 0.0137 1.2110 0.0384 1.0876 0.0161 1.0661 0.0118 λ 99 1.0933 0.0138 1.2558 0.0395 1.1094 0.0161 1.0825 0.0117 λ 00 1.0924 0.0140 1.2548 0.0401 1.1078 0.0163 1.0821 0.0120 λ 01 1.0877 0.0138 1.2536 0.0400 1.1022 0.0159 1.0784 0.0119 λ 02 1.1036 0.0139 1.2794 0.0407 1.1167 0.0159 1.0974 0.0122 λ 03 1.0693 0.0134 1.2222 0.0392 1.0779 0.0153 1.0686 0.0120 λ 04 1.0741 0.0133 1.2361 0.0391 1.0832 0.0149 1.0723 0.0120 λ 05 1.0938 0.0132 1.2694 0.0395 1.1032 0.0147 1.0896 0.0120 λ 06 1.1149 0.0132 1.2980 0.0400 1.1252 0.0147 1.1096 0.0121 Transitory Component ρ 0.6281 0.0629 0.9238 0.0036 0.7261 0.0212 0.2134 0.0210 θ ‐0.3302 0.0439 ‐0.5912 0.0068 ‐0.3717 0.0250 σ2 z 0.1986 0.0153 0.2781 0.0122 0.2243 0.0120 0.1675 0.0136 π 87 1.0000 1.0000 1.0000 1.0000 π 88 1.0592 0.0490 0.9781 0.0378 1.0467 0.0418 1.0711 0.0592 π 89 0.9917 0.0481 0.9201 0.0356 0.9838 0.0402 0.9954 0.0592 π 90 0.9909 0.0453 0.9368 0.0335 0.9875 0.0379 0.9837 0.0556 π 91 0.9703 0.0494 0.9150 0.0372 0.9660 0.0416 0.9678 0.0592 π 92 0.9926 0.0552 0.9285 0.0417 0.9851 0.0467 0.9953 0.0660 π 93 1.0545 0.0459 0.9739 0.0346 1.0366 0.0389 1.0697 0.0557 π 94 1.0213 0.0482 0.9458 0.0368 1.0061 0.0416 1.0301 0.0582 π 95 1.0109 0.0454 0.9186 0.0332 0.9947 0.0385 1.0189 0.0550 π 96 0.9942 0.0436 0.9043 0.0317 0.9755 0.0368 1.0092 0.0528 π 97 0.9481 0.0457 0.8697 0.0333 0.9358 0.0386 0.9551 0.0550 π 98 0.9835 0.0434 0.9011 0.0318 0.9706 0.0365 0.9943 0.0525 π 99 0.9448 0.0433 0.8667 0.0313 0.9371 0.0359 0.9518 0.0520 π 00 0.9558 0.0474 0.8860 0.0350 0.9515 0.0391 0.9591 0.0569 π 01 0.9626 0.0469 0.8925 0.0338 0.9593 0.0387 0.9618 0.0565 π 02 1.0000 0.0488 0.9321 0.0347 1.0012 0.0401 0.9839 0.0599 π 03 1.0771 0.0452 0.9892 0.0322 1.0649 0.0373 1.0755 0.0564 π 04 1.0322 0.0475 0.9495 0.0339 1.0211 0.0393 1.0325 0.0595 π 05 0.9875 0.0404 0.9126 0.0279 0.9855 0.0325 0.9811 0.0500 π 06 The table shows point estimates and standard errors of our error‐components models in equations (2)‐(5). The estimates were obtained by Diagonally Weighted Minimum Distance (see section 4.1). Restricted Model RM1 has no heterogeneous income profiles component. Restricted Model RM2 has no random walk component. Restricted Model RM3 has no MA component.

Table III Persistence of transitory shock (1) (2) (3) (4) Baseline Model Restricted Model Restricted Model Restricted Model periods after shock RM1 RM2 RM3 s 0 1.00 1.00 1.00 1.00 1 0.30 0.33 0.35 0.21 2 0.19 0.31 0.26 0.05 3 0.12 0.28 0.19 0.01 4 0.07 0.26 0.14 0.00 5 0.05 0.24 0.10 0.00 10 0.00 0.16 0.02 0.00 The table shows the fraction of a transitory shock that survives s periods after the shock. Restricted Model RM1 has no heterogeneous income profiles component. Restricted Model RM2 has no random walk component. Restricted Model RM3 has no MA component

Table IV Estimates of Error‐Components Models, Pre‐Tax Household Income Column 1a 1b 2a 2b 3a 3b 4a 4b Male Earnings Sample Household Income Sample Baseline Model Restricted Model RM2 Baseline Model Restricted Model RM2 Parameter Estimate S.E. Estimate S.E. Estimate S.E. Estimate S.E. Permanent Component σ2 α 0.2153 0.0122 0.2121 0.0074 0.2124 0.0107 0.1900 0.0062 σ2 β (x100) 0.1741 0.0370 0.1814 0.0117 0.0799 0.0293 0.1507 0.0105 σ2 γ (x10,000) 0.0103 0.0014 0.0105 0.0009 0.0050 0.0011 0.0069 0.0009 σαβ (x100) ‐0.7891 0.1564 ‐0.7705 0.0825 ‐0.6010 0.1644 ‐0.3256 0.0792 σαγ (x1,000) 0.1556 0.0629 0.1537 0.0376 0.1232 0.0680 0.0250 0.0369 σβγ (x10,000) ‐0.4000 0.0532 ‐0.4110 0.0333 ‐0.2160 0.0401 ‐0.2990 0.0305 σ2 r 0.0014 0.0059 0.0123 0.0048 λ87 1.0000 1.0000 1.0000 λ88 1.0605 0.0166 1.0621 0.0169 1.0466 0.0122 1.0511 0.0135 λ89 1.0786 0.0169 1.0805 0.0173 1.0529 0.0117 1.0580 0.0130 λ90 1.0697 0.0168 1.0718 0.0172 1.0498 0.0117 1.0547 0.0133 λ91 1.1026 0.0180 1.1056 0.0183 1.0617 0.0121 1.0698 0.0139 λ92 1.1095 0.0186 1.1128 0.0189 1.0779 0.0124 1.0880 0.0145 λ93 1.0836 0.0179 1.0864 0.0184 1.0775 0.0123 1.0877 0.0145 λ94 1.0911 0.0184 1.0949 0.0188 1.0795 0.0122 1.0925 0.0146 λ95 1.1129 0.0193 1.1172 0.0195 1.1068 0.0131 1.1230 0.0157 λ96 1.1403 0.0203 1.1458 0.0200 1.1318 0.0135 1.1533 0.0162 λ97 1.1719 0.0217 1.1782 0.0210 1.1525 0.0136 1.1776 0.0163 λ98 1.1581 0.0208 1.1636 0.0207 1.1569 0.0135 1.1815 0.0161 λ99 1.2039 0.0216 1.2104 0.0207 1.1677 0.0133 1.1935 0.0158 λ00 1.1772 0.0210 1.1829 0.0205 1.1626 0.0134 1.1856 0.0158 λ01 1.1295 0.0189 1.1334 0.0191 1.1160 0.0124 1.1283 0.0148 λ02 1.1415 0.0183 1.1452 0.0186 1.1233 0.0121 1.1340 0.0144 λ03 1.1032 0.0174 1.1058 0.0179 1.0984 0.0114 1.1047 0.0134 λ04 1.1175 0.0167 1.1201 0.0172 1.1332 0.0116 1.1416 0.0136 λ05 1.1305 0.0164 1.1327 0.0169 1.1369 0.0115 1.1439 0.0133 λ06 1.1738 0.0170 1.1765 0.0173 1.1390 0.0116 1.1447 0.0133 Transitory Component ρ 0.7403 0.0436 0.7555 0.0222 0.6538 0.0503 0.7540 0.0158 θ ‐0.3531 0.0291 ‐0.3607 0.0276 ‐0.3066 0.0329 ‐0.3475 0.0197 σ2 z 0.1448 0.0127 0.1484 0.0098 0.1582 0.0116 0.1805 0.0084 π87 1.0000 1.0000 1.0000 1.0000 π88 1.0002 0.0587 0.9982 0.0561 0.9783 0.0524 0.9840 0.0434 π89 1.0220 0.0524 1.0201 0.0498 0.9948 0.0459 0.9974 0.0369 π90 0.9936 0.0527 0.9916 0.0503 0.9984 0.0432 1.0012 0.0346 π91 0.9679 0.0550 0.9665 0.0515 0.9822 0.0447 0.9821 0.0360 π92 1.0236 0.0542 1.0200 0.0523 1.0137 0.0476 1.0085 0.0393 π93 1.0689 0.0525 1.0636 0.0508 1.0377 0.0462 1.0303 0.0383 π94 1.0029 0.0515 0.9963 0.0498 0.9969 0.0427 0.9875 0.0352 π95 0.9984 0.0547 0.9940 0.0523 0.9812 0.0481 0.9805 0.0396 π96 0.9488 0.0532 0.9435 0.0498 0.9513 0.0442 0.9498 0.0362 π97 0.9720 0.0570 0.9673 0.0529 0.9990 0.0443 0.9897 0.0358 π98 1.0509 0.0545 1.0463 0.0517 1.0241 0.0439 1.0165 0.0351 π99 0.9867 0.0494 0.9836 0.0450 1.0033 0.0401 0.9976 0.0314 π00 1.0313 0.0530 1.0274 0.0502 1.0339 0.0432 1.0307 0.0346 π01 1.1009 0.0519 1.0974 0.0498 1.1018 0.0428 1.1002 0.0339 π02 1.1185 0.0482 1.1146 0.0462 1.1014 0.0405 1.0989 0.0320 π03 1.1265 0.0486 1.1212 0.0464 1.1405 0.0409 1.1265 0.0317 π04 1.1415 0.0526 1.1361 0.0504 1.1277 0.0430 1.1167 0.0338 π05 1.1357 0.0456 1.1315 0.0435 1.1217 0.0376 1.1134 0.0291 π06 The table shows point estimates and standard errors of our error‐components models in equations (2)‐(5). The estimates were obtained by Diagonally Weighted Minimum Distance (see section 4.1). Restricted Model RM1 has no heterogeneous income profiles component. Restricted Model RM2 has no random walk component. Restricted Model RM3 has no MA component.

Figure I (a) Cross-Sectional Variance (of the Log) by Year 1 0.9 0.8 0.7 0.6 0.5 0.4 Male Earnings Pre-Tax Household Income After-Tax Household Income Figure I (b) Cross-Sectional Gini Coefficient by Year 0.6 0.5 0.4 0.3 Male Earnings Pre-Tax Household Income After-Tax Household Income

Figure II (a) Figure II (b) Decomposition of Cross-Sectional Variance Decomposition of Cross-Sectional Variance Male Earnings, Baseline Model Male Earnings, Restricted Model 1 (no heterog. profiles) 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0 permanent transitory total permanent transitory total Figure II (c) Figure II (d) Decomposition of Cross-Sectional Variance Decomposition of Cross-Sectional Variance Male Earnings, Restricted Model 2 (no random walk) Male Earnings, Restricted Model 3 (no MA component) 0.8 0.8 0.7 0.7 0.6 0.6 0.5 0.5 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0 0.0 permanent transitory total permanent transitory total

Figure III (a) KSS Decomposition of Cross-Sectional Variance Male Earnings 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 permanent transitory total Figure III (b) BPEA Decomposition of Cross-Sectional Variance Male Earnings 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 permanent transitory total

Figure IV Standard Deviation of One-Year and Two-Year Percentage Changes (Volatility) Male Earnings 0.6 0.55 0.5 0.45 0.4 0.35 0.3 One-Year Percent Changes Two-Year Percent Changes

Figure V (a) Decomposition of Cross-Sectional Variance Pre-Tax Household Income, Baseline Model Male Earnings Sample 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 permanent transitory total Figure V (b) Decomposition of Cross-Sectional Variance Pre-Tax Household Income, Restricted Model RM2 (no random walk) Male Earnings Sample 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 permanent transitory total

Figure VI Standard Deviation of One-Year and Two-Year Percentage Changes (Volatility) Pre-Tax Household Income, Male Earnings Sample 0.50 0.45 0.40 0.35 0.30 One-Year Percent Changes Two-Year Percent Changes

Figure VII (a) Decomposition of Cross-Sectional Variance Pre-Tax Household Income, Baseline Model Full Household Sample 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 permanent transitory total Figure VII (b) Decomposition of Cross-Sectional Variance Pre-Tax Household Income, Restricted Model RM2 (no random walk) Full Household Sample 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 permanent transitory total

Figure VIII Decomposition of Cross-Sectional Variance Pre-Tax and After-Tax Household Income, Baseline Model, Male Earnings Sample 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 Permanent, After-Tax Transitory, After-Tax Total, After-Tax Permanent, Pre-Tax Transitory, Pre-Tax Total, Pre-Tax

Table A.1 Age Distribution by Calendar Year Full Household Income Year Male Earnings Sample Sample Mean St Dev Mean St Dev 1987 39 9.9 40 10.0 1988 39 9.8 40 10.0 1989 39 9.8 40 9.9 1990 39 9.7 40 9.9 1991 39 9.7 40 9.8 1992 40 9.7 40 9.9 1993 40 9.6 40 9.8 1994 40 9.6 40 9.8 1995 40 9.6 40 9.9 1996 40 9.6 40 9.8 1997 40 9.6 41 9.8 1998 40 9.7 41 9.8 1999 41 9.6 41 9.8 2000 41 9.7 41 9.8 2001 41 9.7 41 9.9 2002 41 9.7 41 9.9 2003 41 9.7 42 9.9 2004 41 9.8 42 10.0 2005 41 9.9 42 10.1 2006 41 9.9 42 10.1

Figure A.1 Lifecycle Variance Profile of Male Earnings, Controlling for Year Effects Data, Baseline Model, and Restricted Model 2 (no heterogeneous profiles) 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0 5 10 15 20 25 30 35 Data Baseline Model Restricted Model 1 (no heterogeneous profiles)

Figure A.2 (a) KSS Decomposition of Cross-Sectional Variance, P=3 Male Earnings 0.7 0.6 0.5 0.4 permanent 0.3 transitory 0.2 total 0.1 0 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 Figure A.2 (b) KSS Decomposition of Cross-Sectional Variance, P=7 Male Earnings 0.7 0.6 0.5 0.4 permanent 0.3 transitory 0.2 total 0.1 0 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005 Figure A.2 (c) KSS Decomposition of Cross-Sectional Variance, P=9 Male Earnings 0.7 0.6 0.5 0.4 permanent 0.3 transitory 0.2 total 0.1 0 1987 1989 1991 1993 1995 1997 1999 2001 2003 2005

Figure A.3 (a) : Restricted Model 2 Variance Decomposition Male Earnings 0.60 perm 0.40 0.20 tran 0.00 total 1986 1991 1996 2001 2006 Figure A.3 (b) : Restricted Model 2 Variance Decomposition Male Earnings + Spousal Earnings 0.60 perm 0.40 0.20 tran 0.00 total 1986 1991 1996 2001 2006 Figure A.3 (c) : Restricted Model 2 Variance Decomposition Male Earnings + Spousal Earnings + Transfers 0.60 perm 0.40 0.20 tran 0.00 total 1986 1991 1996 2001 2006 Figure A.3 (d) : Restricted Model 2 Variance Decomposition Male Earnings + Spousal Earnings + Transfers + Investment Income 0.60 perm 0.40 0.20 tran 0.00 total 1986 1991 1996 2001 2006 Figure A.3 (e) : Restricted Model 2 Variance Decomposition Total Household Income 0.60 0.40 perm tran 0.20 total 0.00 1986 1991 1996 2001 2006

Figure A.4 (a) Variance Decomposition, Baseline Model Male Earnings DiNardo-Fortin-Lemieux reweighting 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 permanent transitory total Figure A.4 (b) Variance Decomposition, Restricted Model RM2 (no random walk) Pre-Tax Household Income, Full Household Income Sample DiNardo-Fortin-Lemieux reweighting 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 permanent transitory total

Figure A.5 (a) Variance Decomposition, Baseline Model Male Earnings Minimum Threshold: One-Half of Full-Year Full-Time Minimum Wage 0.5 0.4 0.3 0.2 0.1 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 permanent transitory total Figure A.5 (b) Variance Decomposition, Restricted Model 2 (no random walk) Pre-Tax Household Income, Full Household Income Sample Minimum Threshold: One-Half of Full-Year Full-Time Minimum Wage 0.6 0.5 0.4 0.3 0.2 0.1 0 1986 1988 1990 1992 1994 1996 1998 2000 2002 2004 2006 permanent transitory total