PSC. Research Reports. Population Studies Center. Steven J. Haider

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1 Steven J. Haider Earnings Instability and Earnings Inequality of Males in the United States: Report No Research Reports PSC Population Studies Center University of Michigan

2 The Population Studies Center at the University of Michigan is one of the oldest population centers in the United States. Established in 1961 with a grant from the Ford Foundation, the Center has a rich history as the main workplace for an interdisciplinary community of scholars in the field of population studies. Today the Center is supported by a Population Research Center Core Grant from the National Institute of Child Health and Human Development (NICHD) as well as by the University of Michigan, the National Institute on Aging, the Hewlett Foundation, and the Mellon Foundation. PSC Research Reports are prepublication working papers that report on current demographic research conducted by PSC associates and affiliates. The papers are written by the researcher(s) for timely dissemination of their findings and are often later submitted for publication in scholarly journals. The PSC Research Report Series was begun in 1981 and is organized chronologically. Copyrights are held by the authors. Readers may freely quote from, copy, and distribute this work as long as the copyright holder and PSC are properly acknowledged and the original work is not altered. PSC Publications Population Studies Center, University of Michigan S. University, Ann Arbor, MI USA

3 Earnings Instability and Earnings Inequality of Males in the United States: by Steven J. Haider Research Report No July 1997 Abstract: Although much recent research has focused on changes in the distribution of annual earnings, much less research has examined changes in the distribution of lifetime earnings. Have lifetime earnings become more unequal across individuals? Have earnings become more unstable over the life-cycle for individuals? Using both a simple descriptive analysis and a parametric analysis with the Panel Study of Income Dynamics, I answer these questions for the United States for the period The main results include: (1) lifetime earnings inequality increased substantially during the early 198 s, (2) earnings instability increased during the 197 s and is counter-cyclical, and (3) annual inequality increased because of fairly equal increases of a persistent component and an instability component. Dataset used: Panel Study of Income Dynamics This paper has benefited greatly from extensive discussions with Gary Solon, Deborah Reed, and Shinichi Sakata, as well as from comments from the participants of the University of Michigan s Labor Seminar and Graduate Student Seminar. Computing resources were provided by the Population Studies Center at the University of Michigan. An earlier draft of this paper was presented at the 1997 Meetings of the Population Association of America, Washington, DC. Department of Economics, University of Michigan, Ann Arbor, MI 4819; sjhaider@umich.edu.

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5 1. Introduction A vast literature has emerged which documents substantial changes in annual earnings inequality in the United States. After remaining relatively stable from the 194's to the 197's, annual earnings inequality increased gradually during the 197's and increased rapidly during the 198's. 1 Although these trends in annual inequality indicate important changes in the economy, they provide little information about lifetime earnings inequality [see Lillard (1977)]. Inferences about lifetime inequality cannot be drawn from repeated measures of annual inequality because there are substantial movements through the earnings distribution and the degree of movement can change. In many respects economists will be more concerned about changes in lifetime inequality than in annual inequality. Lifetime inequality measures the disparity in the long-term resources of individuals and thus ignores earnings dierences in one year which are compensated in another year. Economists have long recognized that optimizing agents will attempt to smooth consumption over such earnings uctuations [see Friedman (1957)]. Despite the signicant changes in annual inequality during the last two decades, very few studies directly measure recent changes in lifetime (or long run) earnings inequality. 2 The lack of such research is not surprising given its substantial data requirements. A few recent studies infer changes in lifetime earnings inequality by considering annual inequality and earnings mobility jointly, where earnings mobility refers to the movement of individuals through the earnings distribution over time. 3 To make appropriate inferences about the relationship between mobility, annual inequality, and lifetime inequality, mobility must be evaluated within a suciently rich framework to capture very dierent types of movements through the earnings distribution. For example, an individual who has low initial earnings may systematically move up through the distribution because of a relatively high growth rate; such compensatory earnings growth could be due to dierential human capital investments. The eect of an increase in such mobilitymay increase or decrease lifetime earnings inequality depending on the initial position of such an individual in the lifetime earnings distribution. Alternatively, an individual may be mobile through the earnings distribution simply because of transitory earnings shocks, or earnings instability. Such instability will not appreciably change lifetime earnings inequality if the shocks are suciently transitory. Drawing appropriate conclusions about lifetime earnings inequality requires distinguishing between these dierent types of mobility. Regardless of its implications for lifetime earnings inequality, changing earnings instability is interesting in its own right but it has been studied by only a few researchers [see Gottschalk and Mott (1994) and Mott and Gottschalk (1995)]. With realistic assumptions such as the existence of liquidity constraints, risk aversion, and uncertainty 1 See Blinder (198) and Karoly (1993) for a historical perspective, and see Levy and Murnane (1992) for a review of recent studies. 2 One notable exception is Gottschalk and Mott (1994) who calculate the change in long run (ageadjusted) earnings inequality for the periods and Studies which infer changes in long run inequality from recent changes in annual inequality and mobility are Gittleman and Joyce (1996), Buchinsky and Hunt (1996), and Mott and Gottschalk (1995) : instab.tex 3

6 about one's future earnings stream, it is straightforward to write down a consumer choice model where increases in earnings instability are welfare reducing for an individual; quite simply, earnings instability will hinder consumption planning and the ability to smooth consumption. 4 One might suspect that earnings are becoming more unstable because of various secular trends discussed in dierent economics literatures and speculated upon in the popular press. For example, there seems to be popular sentiment that there are increasing levels of layos and job separations. In a series of articles in the New York Times reproduced into the volume The Downsizing of America, a collection of analyses, interviews, and opinion polls leave the reader with the strong impression that involuntary job separations are becoming increasingly common. 5 In addition, a number of studies have found that there have been dramatic increases in the use of temporary workers and contingent workers [see Segal and Sullivan (1997)]. One avenue through which both of these trends will be welfare reducing and observable is through increasing the instability of earnings. In this paper, I examine changes in lifetime earnings inequality and earnings instability for males using 25 years of the Panel Study of Income Dynamics (PSID). I examine only males because of the large changes in female labor force participation during the relevant period even though this restriction certainly overlooks interesting dierences in the economy. 6 I rst examine long run earnings inequality by comparing the distribution of long run earnings measured over ten years for two time periods. I nd that long run earnings have become more unequal from the 197's to the 198's, but there are many reasons to nd this answer unsatisfactory. To overcome some of the drawbacks, I model the process that determines lifetime earnings and then measure lifetime earnings inequality based upon the model. I obtain answers with the parametric method which are consistent with the descriptive method, and I am able to describe more precisely changes in the receipt of lifetime earnings. My ndings include that most of the changes in lifetime earnings inequality occurred during the early 198's. In addition, earnings instability increased primarily during the 197's and is counter-cyclical. Finally, I examine the implications of the model for annual earnings inequality. Increases in an instability component and a persistent component are equally important to explaining the increases in annual earnings inequality over the entire period, but increases in the instability component were more important during the 197's and increases in the persistent component were more important during the 198's. Finally, annual inequality is largely counter-cyclical because earnings instability iscounter-cyclical. 4 For example, see Zeldes (1989) for empirical evidence that liquidity constraints aect consumption in the United States. 5 Whether this perception is true is debatable because studies using dierent large-scale data sets have come to dierent conclusions. Studies using the CPS [Farber (1996), Diebold et al. (1997)] nd that job tenure has remained fairly constant while studies using the PSID [Mott and Gottschalk (1994)] nd that job tenure is decreasing for white males. This dierence does cast some doubt as to whether the ndings of earnings instability in this paper based upon the PSID would be robust across data sets. 6 Although Gittleman and Joyce (1996) nd long run inequality to be fairly constant for men during the period , they nd signicant declines in long run inequality for women : instab.tex 4

7 A few recent studies have examined lifetime earnings inequality (or long run inequality) and earnings instability. Gittleman and Joyce (1996) examine long run inequality with 25 years of matched data from the Current Population Survey (CPS). Although the matched CPS has the advantages of having a larger sample size, including the ability to examine dierent demographic groups, it only has two years of data for any one individual. Consequently, they are not able to separate earnings instability from more general earnings mobility, and their ability tomake inferences about long run inequality, even for ve years, is limited. Mott and Gottschalk (1995), using the PSID, focus on changes in the parameters of the autocovariance structure of earnings for within-group inequality (groups dened by education and age). Buchinsky and Hunt (1996) infer changes in lifetime inequality by considering earnings mobility and annual inequality jointly with the National Longitudinal Survey of Youth (NLSY). Although their study has the benet of considering more demographic groups, the cohort nature of the NLSY makes it dicult to separate life-cycle and time eects. Two recent studies examine earnings instability. Gottschalk and Mott (1994) measure earnings instability with a much simpler model which is fairly restrictive with respect to the temporal changes allowed. Mott and Gottschalk (1995) measure earnings instability with a more realistic model than their 1994 study, and their results will provide an interesting comparison to my results. The rest of the paper is organized as follows. I present a simple, descriptive analysis of changes in the distribution of long run earnings in Section 2. Section 3 provides the details of the heterogeneous growth model used for the parametric analysis and derives the implications of the model for earnings inequality and earnings instability. Section 4 provides details of the sample selection criteria, variable denitions, and the estimation procedure. I present the empirical results in Section 5 and conclude with Section 6. Details of the estimation procedure, various derivations, and sample selection are left to the appendices. 2. A Descriptive Analysis of Lifetime Earnings Inequality Has inequality in lifetime earnings changed during the last three decades? A simple, descriptive method of answering this question is to construct the distribution of lifetime earnings for individuals from separate birth cohorts. The distributions can then be compared for evidence of increased inequality. The advantages of such a method are that it is simple and no assumptions are necessary regarding how earnings are received. A data set well-equipped to perform such an analysis is the PSID because it has earnings information for individuals from the years In particular, I use the Survey Research Center (SRC) subsample of the PSID, Waves I through XXV, and I use the eld \Total Wages and Other Labor Income" as my earnings measure, adjusted to 1994 dollars by the CPI-U-X1. 7 Even though this panel length is unrivaled for multi-cohort data sets, further compromises must be made in measuring lifetime earnings inequality because 7 The SRC subsample is a nationally representative subsample of the PSID, so weights are not used : instab.tex 5

8 the panel is still short relative to a lifetime. I proxy lifetime earnings with the present discounted value (at 5%) of ten years of real earnings for prime-aged individuals, and I compare the distributions for the years and ; I refer to the approximation as long run earnings. Finally, I restrict the cohorts for each period to male household heads who are between the ages of 3 and 44 in the rst year. 8 To describe the distributions, I calculate various quantiles, the 9th/1th and 8th/2th quantile ratios, the coecient of variation, and the variance of log earnings. In Table 1, I present the results of the calculations for four samples: all males, males with positive earnings in every year, white males, and white males with positive earnings in every year. Various trends are readily apparent in Table 1. First, the lower quantiles decline and the upper quantiles increase between the two time periods for each sample, implying that the long run earnings distribution is spreading out. For example, males with positive earnings in every year (positive earning males) in the top decile enjoyed a 1% increase in long run earnings, but similar earners in the bottom decile suered a 9% decline. However, the magnitudes of the quantiles are highly dependent on the price deator used and thus should be interpreted with caution. 9 Quantile ratios, direct measures of the spread of a distribution, are not as deator dependent and increase consistently between the time periods for each sample. 1 The other reported measures of inequality, the coecient of variation and the variance of log-earnings, also increase between the time periods, with the variance of log-earnings increasing 31% and 35% for positive earning males and white positive earning males respectively. Thus, long run inequality appears to have increased between 1969 and Comparing the various samples, the restriction to positive earning males serves to dampen changes in the quantile ratios but the restriction to white positive earning males does not. Presumably, the restriction to positive earning males dampens changes in inequality because this group is selected ex post to have relatively stable earnings; namely, they must not have any unemployment spells which encompass an entire calendar year. Although this restriction dampens the measured change in inequality, positive earning males are the vast majority of the sample, representing 97% and 92% of all males for the periods and respectively. 11 The restriction to white males does not change the results because there are very few non-whites in the PSID-SRC subsample. For example, only 52 of all males (6.6% of the sample) are non-white for the period I include all individuals who have a complete set of observations for the ten year period. I consider any earnings observation which is a \major assignment" to be missing. 9 In particular, if the deator over-compensates for the rate of ination, the real growth rate in earnings will be underestimated. Such an error will cause (1) the growth rate of long run earnings between the two periods to be biased downward and (2) too much weight tobegiven to the early earnings years in the PDV calculation. 1 Quantile ratios will only be deator dependent to the degree that ination is over-estimated dierently over time. 11 Presumably, this increase in the number of individuals with zero-earnings during a calendar year contributes to the higher inequality for the sample including all individuals. For further evidence that employment rates are lower for those who tend to have lower earnings, see Gottschalk (1997) : instab.tex 6

9 Although the results reported above are certainly suggestive, there are good reasons to nd the results inadequate in providing a clear picture of changes in lifetime inequality. The age (3-44 in the rst year) and time horizon (1 years) restrictions immediately beg the question of whether all compensatory earnings dierences have been ignored; to the degree that high earning, prime-aged individuals are being compensated for low initial earnings, perhaps because of education and training, important life-cycle reasons for inequality exist in the long run earnings measure used. In addition, such a comparison masks the underlying characteristics of any change. For example, did the change in the process that determines lifetime earnings happen smoothly over the period? Or was there a sudden regime shift? The answers to these questions will be important in predicting what will happen in the future and focusing the search for potential causes. Third, the methodology ignores potentially useful information because it requires the calculation of long run earnings to calculate distribution summary statistics. Such a method does not provide a means to include the information provided by an individual observed less than a lifetime (or the time period chosen to proxy for a lifetime). Such \partial" earnings information is important to understanding recent changes in lifetime inequality. Parametric methods can also be used to examine questions about the distribution of lifetime earnings and have several advantages over the descriptive analysis above. For example, a exible, parametric specication can help to pin down the character of change in lifetime earnings inequality. In addition, parametric methods can provide the structure necessary to incorporate information from individuals for whom only a \partial" set of observations are available. Of course, parametric methods suer the signicant disadvantage that any answers which are obtained are reasonable only to the extent that the parametric specication captures the underlying process which generates lifetime earnings. The focus of the rest of the paper is to further examine lifetime earnings in a parametric framework. 3. A Parametric Analysis of Inequality and Instability In the following parametric analysis, I model the underlying process of earnings and then derive the implications of the model for the questions in which I am interested. In particular, I specify a model which will be informative with respect to changes over time in lifetime earnings inequality and earnings instability. 3.1 A Model of Heterogeneous Growth Researchers have frequently used a heterogeneous growth model to describe the underlying process of life-cycle earnings. 12 The intuition for the model is that each individual can have a unique life-cycle earnings prole, although the heterogeneity in proles is limited by functional form restrictions. The functional form restrictions are usually imposed by specifying individual proles in deviation form. For example, a standard specication for a heterogeneous growth model is, log[y it ]=f(x it ; t )+( i + i X it )+" it ; (1) 12 For example, see Baker (1997), Mott and Gottschalk (1995), MaCurdy (1982), Hause (198), and Lillard and Weiss (1979) : instab.tex 7

10 where Y it is real earnings, X it is (potential) experience, f(:) is a polynomial function of experience, and " it is a mean-reverting earnings shock. The inclusion of the heterogeneous growth term ( i + i X it ) allows individuals to have dierentintercepts (see Figure 1), slopes (see Figure 2), or any combination of the two (see Figure 3), relative to the mean prole f(:). Such a parameterization restricts the change in slope (the second derivative with respect to experience) to be equal across individuals; since the model is specied with logearnings, this assumption is equivalent to assuming that the percentage change in earnings growth is equal across individuals. Finally, most previous researchers assume that the variance of individual proles around the mean prole is constant over time (var[ i ]=, 2 var[ i ] = 2, and cov[ i; i ] = ) and that the transitory shocks are uncorrelated with the individual specic parameters (cov[ i ;" it ] = cov[ i ;" it ] = ). A heterogeneous growth model is intuitively appealing because it seems likely that individuals will dier systematically from the mean earnings prole, particularly over long horizons, and it will have clear implications to lifetime earnings inequality and earnings instability. 13 There are many theoretical justications for a model of heterogeneous earnings growth. In fact, appending heterogeneity to just about any model which predicts an upward sloping earnings prole will provide a compelling justication. For example, heterogeneity in human capital investment across individuals with similar ability [e.g. Becker (1975)] or heterogeneity across rms in monitoring costs [e.g. Lazear (1981)] will imply a model of heterogeneous earnings growth. Regardless of why individuals have dierent proles around the mean prole, equation (1) will capture any such heterogeneity. The specication in (1) is restrictive for my purposes because the distribution of individual proles relative to the mean prole cannot change over time. This restriction is undesirable because it precludes any changes in the process that determines relative lifetime earnings, and therefore precludes any changes in relative lifetime earnings inequality. To loosen this restriction, I include a period specic factor loading which allows an individual's deviation from the mean prole to change. If the heterogeneous growth term ( i + i X it )is determined by skill, perhaps obtained through dierential human capital investment, then the factor loading may be interpreted as the relative price of skill. Specically, suppose life-cycle earnings follow the process, log[y it ]=f(x it ; t )+p t ( i + i X it )+" it : (2) The factor loading p t is necessarily normalized to equal one in the rst period. As p t increases (decreases), the factor loading serves to accentuate (attenuate) any individual deviations from the mean prole. If p t were zero, individuals would only deviate from 13 A common alternative specication for life-cycle earnings is a random walk model [see Mott and Gottschalk (1995) and Abowd and Card (1989)]. Several studies have empirically compared the two models but the results do not indicate either model to be denitively superior [see Baker (1997), Abowd and Card (1989), and MaCurdy (1982)]. The lack of a clear empirical choice is not surprising given that the covariance structure of the models are only statistically distinguishable by functional form restrictions and the models are not theoretically mutually exclusive. Considering the statistical similarities and the suitability of a heterogeneous growth model to the questions I wish to ask, I focus on a heterogeneous growth model : instab.tex 8

11 the mean prole because of the mean-reverting, possibly heteroskedastic error term " it. In addition, I specify the mean prole f(:) to be a quartic in experience rather than a quadratic in experience. 14 Although the parameterization in (2) is fairly simple, it does allow for year-to-year changes in the process that generates relative lifetime earnings. Equation (2) still restricts the higher order derivatives to be equal across individuals. Because I specify f(:) tobe a quartic, this restriction is desirable because an unrestricted quartic could t many proles which would not be considered stable (a restriction necessary below); for example, a trough in earnings at middle age could conceivably be consistent with an unrestricted quatric. In addition, I retain the assumption that the variances of the individual parameters do not change over time (var[ i ] =, 2 var[ i ] = 2, and cov[ i ; i ] = ). However, because of the introduction of the factor loading p t, this assumption implies that the relative variances of individual proles are constant, where previously, the assumption implied that the absolute variance in proles remained constant. Finally, Iretain the assumptions that transitory shocks and the individual specic parameters are uncorrelated, cov[ i ;" it ]=cov[ i ;" it ]=. 3.2 Lifetime Earnings Inequality To derive the implications of a heterogeneous growth model for lifetime earning inequality, I rst dene lifetime earnings. Let lifetime earnings for i in t (denoted as Y it )be the present discounted value of earnings for individual i if he worked from experience level to R (retirement) at the parameter values of year t, Y it = Z R exp[,rx] exp[f(x; t )+p t ( i + i X)] dx: (3) Intuitively, I am simply summing under an individual's earnings prole for a given year, discounting appropriately. There are two important aspects to this denition of lifetime earnings. First, since no individual actually receives a lifetime's worth of earnings in the conditions of a particular year, such a measure should be considered \projected" lifetime earnings or the econometrician's \expected" lifetime earnings. Such a measure is useful because it tracks year-to-year changes in the underlying process of earnings and the magnitude has a clear interpretation with respect to lifetime earnings. Second, this denition of lifetime earnings ignores the transitory component (" it ) because " it are mean-zero transitory shocks which do not change an individual's lifetime earnings prole and will thus have little impact on the level of lifetime earnings. 15 An alternate justication for ignoring transitory shocks is 14 Murphy and Welch (199) nd that specifying f(:) asaquadratic imposes growth rates which are too low for the earliest and latest parts and too high for the middle part of the life-cycle relative to the empirical earnings prole. They further conclude that the further exibility of a quartic ts quite well. 15 One can show that the variance induced in lifetime earnings by realizations of "it approaches zero as the time horizon of working becomes larger and larger. A simple proof of this claim is presented in Appendix B : instab.tex 9

12 that transitory shocks have little eect on the consumption decisions of individuals under uncertainty [see Friedman (1957)]. But for such a justication to be valid, the earnings process in (2) would need to coincide with the earnings expectations of individuals. Given the assumptions made thus far, the variance of the logarithm of lifetime earnings may beshown to be [the derivation is in Appendix B] var[log Y it] _= p 2 t ( "R R h X exp[f(x; t )] dx 2 + R R exp[f(x; t)]dx "R R h i Xexp[f(X; t )] dx +2 exp[f(x; t)]dx R R i # ) # 2 2 : (4) To derive the expression in (4), I approximate the variance of log-lifetime earnings with the variance of a Taylor Series approximation of log-lifetime earnings in the neighborhood of E[ i ]=. The expression inside the large square brackets represents the marginal impact of a change in i on the logarithm of lifetime earnings near E[ i ]. Since i is estimated to have a small variance around its mean both in previous studies and this study, this approximation should be fairly good. 16 Akey aspect of the measure of lifetime inequality in (4) is that it is not a function of the individual specic parameters ( i and i ), but rather it is a function of the variance of the individual specic parameters ( 2, 2, and ). Consequently, I do not need to estimate the earnings prole for each individual, thus allowing the inclusion of individuals observed even for a single year. Furthermore, all of the parameters in (4) are directly estimable as part of the autocovariance structure of earnings. The intuition for why this is the case is that the autocovariance describes how earnings tend to evolve over time; by knowing the earnings distribution in various years and by knowing how earnings tend to evolve, I can back out estimates of lifetime earnings inequality. The time series of (4) will be reported as the level of lifetime earnings inequality in each year. 3.3 Earnings Instability I dene earnings instability as temporary deviations from one's earnings prole. 17 Such a denition attempts to separate compensatory earnings growth, perhaps because of dierential human capital investment, from transitory earnings shocks due to volatility in the economy. Although theoretically I would want to dene instability as deviations from an individual's expected earnings to retain a \surprise" interpretation, good information on individual expectations is not generally available. In addition, if earnings were found to 16 See results below, and see Baker (1997), and Hause (198) for studies in which 2 is estimated to be small. 17 Because I will specify "it to follow an ARMA process, not all parameter values are consistent with a temporary interpretation. However, the parameter estimates below imply that less than 15% of a transitory shock remains after 3 years, thus providing support for the interpretation to follow : instab.tex 1

13 become more unstable with respect to an individual's expectations, the researcher could not be certain if the formation of individual expectations changed or if the volatility of earnings receipt changed. Therefore, to the degree that a heterogeneous growth model is similar to how individuals form expectations of life-cycle earnings, instability may be interpreted as unexpected earnings shocks and appropriate welfare conclusions may be drawn. Even if the coincidence does not occur, I am forming the measure of instability consistently over time and any changes I nd will represent real changes in how earnings are received. From equation (2), the deviation of individual i in period t from his prole is measured by the term " it. To measure the magnitude of earnings instability inyear t, I use the crosssectional variance of the deviations in year t, 2 " t. I allow an individual's deviation from his prole to be serially correlated by modeling " it as a low-order ARMA(p,q) process. Consider the case of an ARMA(1,1), " it = " i(t,1) + i(t,1) + it : (5) The primitive error term it is the individual/time-specic transitory shock and the variance of the primitive error term ( 2 t ) measures the contemporaneous volatility in the economy. Because of the \memory" inherent in an ARMA process, the transitory shocks it may build up over time in " it, the actual distance an individual is from his prole. I allow the variance of the primitive error term ( 2 t ) to vary over time. One important caveat to interpreting " 2 t as earnings instability is that " 2 t will also capture any mean-reverting measurement error in the earnings data. Thus, " 2 t will overestimate the variance of mean-reverting earnings shocks in the economy for a given year. Although measurement error is problematic in interpreting the levels of " 2 t, there is little reason to believe measurement error will aect the interpretation of the changes in " 2 t. 18 Since the primary focus of the paper will be changes in earnings instability, measurement error will be ignored in the rest of the paper. Given the manner in which I parameterize earnings proles, there exists another avenue through which earnings can uctuate from year to year for individuals. Recall that movements of p t serve to accentuate and attenuate individual specic earnings proles as p t increases and decreases respectively. Changes in the volatility of p t will thus change the amount of year to year earnings uctuations in the economy. I will examine the time series plot of p t for any evidence of increased volatility. 18 Because I nd that "t 2 increased over time, respondents would need to report earnings with more measurement error over time to cause the observed increase. Duncan, Juster, and Morgan (1983) argue that panel data will be measured with less error over time, particularly for economic phenomena, because respondents \get trained to their task, and cross-year consistency checks are applied." Thus, the increases in instability may actually be underestimated if measurement error has declined. In addition to respondent error, measurement error could increase over time in the PSID due to reductions in the editing budget and due to increased use of proxy respondents. According to conversations with Tecla Loup of the PSID sta, signicant reductions in the editing budget began with the 1993 wave, thus budgetary changes are not of concern for the sample in this paper. Bound and Krueger (1991) nd that proxy-reported earnings in the CPS are comparable in accuracy to self-reported earnings; there is little reason to expect the PSID would be dierent : instab.tex 11

14 3.4 Annual Earnings Inequality A heterogeneous growth model captures a few insights concerning annual earnings inequality that other researchers have explored. Given the assumptions made so far, the heterogeneous growth model in (2) implies that the inequality of annual experience-adjusted earnings is var[ log Y it jx it ]=p 2 t( X2 it +2 X it )+ 2 " t : (6) From this expression, it is clear that increases in earnings instability ( 2 " t ) will cause crosssectional earnings inequality to increase, thus a model of heterogeneous growth captures the central insight of Mott and Gottschalk (1994, 1995): observed changes in cross-sectional earnings inequality may be due to changes either in earnings instability or \persistent" inequality. 19 I provide estimates as to the importance of instability and persistent inequality to annual experience-adjusted inequality. How does mobility enter a model of heterogeneous growth? Mobility of a transitory nature (earnings instability) enters the model through " it. Realizations of " it may cause individuals to switch places in the earnings distribution, but such mobility will have little impact on lifetime earnings inequality. 2 Mobility of a compensatory nature enters the model through,, 2 and 2.21 The term will be of primary importance because it measures the propensity of individuals who begin lower (higher) in the distribution to have relatively higher (lower) earnings growth rates. Compensatory mobility will also be determined by 2 because this parameter measures the initial spread of earners and by 2 because this parameter measures the spread of possible growth rates; both parameters will aect the possibility of individuals being able to change positions in the distribution. Changes in compensatory mobility could have a large impact on lifetime earnings inequality. Finally, a standard human capital model predicts that earnings inequality should rst decrease then increase over the life-cycle for a given birth cohort. 22 Such a prediction comes from the observation that individuals who invest in more human capital will tend to have lower initial earnings as they undertake human capital investment, but such individuals tend to be rewarded with higher earnings growth rates. Simply taking the derivative of (6) with respect to potential log Y it jx it it =2p 2 t( 2 X it + ): (7) For a negative as many studies have found (including this study), it is clear that such a trend is likely. Thus, experience-adjusted annual inequality is dependent on the level of 19 I refer to the rst term of the right hand side of (6) as persistent inequality because changes in these parameters aect lifetime earnings inequality. 2 See Appendix B. 21 Because of the ndings in this paper and that of many others, the following discussion assumes that is negative. 22 See Mincer (1974) for a theoretical exposition and Hause (198) for an empirical study : instab.tex 12

15 experience at which inequality is being evaluated and unadjusted inequality is dependent on the distribution of experience in the economy. Such considerations cast doubt on temporal conclusions drawn about inequality from data sets in which the distribution of experience changes systemically over time, such as in cohort or balanced panel data sets. 4. The Estimation Procedure 4.1 The Data and Panel Construction To estimate the parametric model, I use the same data set used for the descriptive analysis, the SRC subsample of the PSID, Waves I-XXV. I use the same annual earnings measure as before, \Total wages and other labor income," adjusted to 1994 dollars using the CPI-U-X1. I further restrict my analysis to male household heads who are white, who are not students or retirees, who have no observed years of zero earnings, who are aged 25-6, and who are observed at least twice. I restrict the sample to whites because there is good reason to believe that the labor market experience was dierent for non-whites than whites during the sample period, but the small number of non-whites in the SRC is not sucient for separate analysis [see Table 1]. 23 The restriction to individuals with no years of zero earnings is conservative in that this group will tend to have less earnings variance than a group including all earners because the group is selected ex post to have relatively stable employment; even so, the restriction aects very few of the men in the sample because of the high labor force attachment of males [see Table 1]. Further details about the sample restrictions are in Appendix C. With these sample restrictions, I construct a revolving balanced panel from the data in which participants are required to have positive earnings only for the years for which they satisfy the other sample restrictions, such as the age range, student status, and retirement status. For example, a person aged 5 in 1968 will be kept in the sample until he reaches age 6 in 1977, or until he retires, although no data are present at least for the years This type of panel design diers from a standard balanced panel design in which a person aged 5 in 1968 would not be kept in the sample because there will be at least 15 years of missing data, and it diers from an unbalanced panel design in which every positive earnings observation for every individual is included, regardless of whether there are intervening years with zero earnings. The sample restrictions leave a sample size of 3,115 individuals and 35,73 person-year observations. Summary statistics for this primary sample are available in Table 2. The structure of the panel necessitates statistical procedures to handle missing data, detailed in Appendix A. 23 With these restrictions, I am certainly overlooking some of the most interesting comparisons but they are necessary given the data set. Not only are the earnings dynamics of white males well-studied already, but Gittleman and Joyce (1996) and Marcotte (1995) nd important dierences between white males and other demographic groups : instab.tex 13

16 4.2 Estimation As alluded to previously, all of the parameters of interest in (4), (5), and (6) are estimated as part of the autocovariance structure of earnings. I estimate the parameters of the autocovariance structure in two stages. In the rst stage, I estimate with OLS the parameters t of the mean prole f(:) and the residuals y it from the equation, log Y it = f(x it ; t )+y it : (8) From the residuals y it, I calculate the individual P empirical covariance matrix C i and the mean empirical covariance matrix C = n,1 i C i. I present the empirical covariance matrix in Table 3, along with standard errors and sample sizes, for a rst stage regression with time varying rst stage parameters ( t ) and a quartic in experience. I also present results below for a rst stage regression with the coecients restricted to be equal across years (except for the intercept), but I do not present the corresponding Table 3 in the paper. 24 The empirical autocovariance matrix in Table 3 is consistent with those calculated by other researchers. The variance of adjusted log-earnings does increase over time (looking down the principal diagonal) with the trend accelerating during the 198's. The elements o the principal diagonal are positive and signicantly dierent from zero, implying that earnings deviations are positively, serially correlated. In addition, the autocovariances decline over time (looking down columns), consistent with a stationary covariance process such as an ARMA(p,q). Finally, the magnitudes in the matrix are very similar to the matrices constructed by other researchers [see Mott and Gottschalk (1995) and Baker (1997)]. The empirical covariance matrices are the basis of my analysis in that I t a variety of covariance structures to these estimates. I estimate the parameters of the autocovariance structure in a second stage using a Generalized Method of Moments framework. The model in (2) implies that the theoretical covariance matrix (X i ; t ) has the typical diagonal element var[y it jx i ]=p 2 t( X2 it +2 X it )+ 2 " t ; which is a restatement of (6), and the typical o-diagonal element cov[y is ;y it jx i ]=p s p t [ (X is X it )+ (X is + X it )] + "s " t : (9) Let the vector c i be the unique elements of the empirical covariance matrix C i and let the vector (X i ; t ) be the unique elements of (X i ; t ), stacked conformably with c i. The 24 As a further check of the robustness of the results, I estimated all the models in the paper with the rst stage containing only a quadratic in experience. None of the substantive conclusions changed, so for brevity, I only report the results using a quartic in the rst stage : instab.tex 14

17 moment conditions which are used to identify the parameters are E[c i, (X i ; t )] = : The moment conditions imply that I choose the parameters of the theoretical covariance matrix (X i ; t ) so that (X i ; t ) is as close as possible to the empirical covariance matrix C. Due to the ndings of other researchers, I use the identity matrix as the weighting matrix rather an optimal weighting matrix. 25 The identity matrix will provide consistent but not asymptotically ecient estimates in general. Appendix A provides further details on how I employ the estimation procedure with the revolving balanced panel design. I present estimates using two specications for the process of the transitory earnings shock " it, an ARMA(1,) and an ARMA(1,1), and two specications for the rst stage regressions, time varying coecients on the polynomial of experience and time invariant coecients on the polynomial of experience (but with year specic intercepts). 26 In every specication, I allow the underlying error term in the ARMA(p,q) process to be heteroskedastic across time. I refer to these specications with the labels, (I) " i;t follows an ARMA(1,) and time varying rst stage coecients, (II) " i;t follows an ARMA(1,1) and time varying rst stage coecients, (III) " i;t follows an ARMA(1,) and time invariant rst stage coecients, and (IV) " i;t follows an ARMA(1,1) and time invariant rst stage coecients. Specication II is the preferred statistical model in that it is the most exible, but all specications will be presented in the graphs to demonstrate the robustness of the reported results. 5. Results The second stage parameter estimates and standard errors for each of the specications are presented in Table 4. To compare the performance of the second stage models, I calculate the test statistic suggested by Newey (1985). All of the parameters are estimated with the expected sign and reasonably precisely, with t-statistics ranging from approximately 2 to 1. The parameters which have comparable counterparts in other studies are of similar magnitudes. Notably, the covariance of the individual parameters ( ) is negative as other studies have found, suggesting that there are compensatory dierences in earnings [See Baker (1997) and Hause (198)]. In addition, the MA(1) parameter is signicantly dierent from zero for both rst stage specications, suggesting models II and 25 Altonji and Segal (1996) nd that the Optimal Minimum Distance estimator which uses the estimated covariance matrix of the moments as the weighting matrix has poor small sample properties. Following the suggestion of Altonji and Segal (1996) and the practice of other researchers in this area, I use the identity matrix as the weighting matrix. 26 Baker (1995) and Mott and Gottschalk (1995) nd that an ARMA(1,1) is sucient to t the mean reverting process in earnings data. Mott and Gottschalk (1995) nd evidence that the memory parameter of the AR(1) process increases over time, but I restrict it to be constantinmy specication : instab.tex 15

18 IV should be preferred to models I and III. It is worth noting that the t of the second stage model, as well as the parameter estimates, is very similar between the two rst stage specications (comparing specications I vs. III and II vs. IV); although comparing the second stage goodness of t statistics is not an appropriate basis for rst stage model selection, the comparison does imply that the autocovariance structure of the residuals is very similar whether or not the rst stage time constraints are imposed. 27 The focus of this paper is not the parameter estimates in Table 4, but the implications of the parameter estimates for lifetime earnings inequality, earnings instability, and annual earnings inequality. Using the formulas derived in Section 3, I calculate measures for each of these phenomena. 5.1 Estimates of Lifetime Earnings Inequality As a benchmark to interpret my results, I rst present experience-adjusted annual earnings inequality in columns 1 and 2 of Table 5, and I graph the estimates in Figure 4. The estimates for specications I and II are simply the principal diagonal elements from the empirical covariance matrix in Table 3, and the estimates for specications III and IV are calculated similarly. For both of the rst stage specications, experience-adjusted annual earnings inequality increases approximately 7% from 1967 to These results are very similar to those found by previous researchers. For example, using the March CPS, Juhn, Murphy, and Pierce (1993) report that earnings inequality increased about 8% for the period using a similar inequality measure. To examine changes in lifetime earnings inequality, I calculate the time series implied by equation (4). I present the calculations for specications II and IV in Table 5, and I graph the results for all four specications in Figure 5. Consistent with the ndings of others, lifetime inequality is less than experience-adjusted earnings inequality in each period [See Lillard (1977)], which is what one would expect given the existence of compensatory earnings growth ( is negative). For each specication, lifetime inequality declined slightly during the 197's and then increased substantially during the 198's. This trend represents an increase in lifetime inequality of 79% for specication II, an increase 27 An appropriate rst stage model selection procedure is to perform an F-test on whether the time invariant specications (III and IV) are restrictive when compared to the time varying specications (I and II). Given the large sample size and that the time invariant model implies that 96 restrictions are being jointly imposed, such an F-test not surprisingly rejects the imposition of the time invariant restrictions. Perhaps more telling of the similarity in t between the two models is that the unadjusted R-square increases from.389 to.443 when moving from 29 (the time invariant specication) to 125 parameters (the time varying specication). 28 I calculate increases of 73% and 67% for specications I/II and III/IV respectively by comparing the rst three years and last three years of estimates. The spike in the nal year of the sample is robust across various other sample restrictions, but I still compare the averages of the rst three years and last three years so that the spike does not overly inuence the estimated increase. If I do compare between just the rst year and the last year, the increase is 1% and 92% for specications I/II and III/IV respectively : instab.tex 16

19 comparable to the increase in experience-adjusted annual inequality. 29 For specication II, the change in lifetime inequality may be interpreted as follows: an individual with lifetime earnings one standard deviation above the mean would have had lifetime earnings 37% above the 1967 mean, but a similar individual in 1991 would have had lifetime earnings 53% above the 1991 mean. The results from the descriptive and parametric analyses are consistent in indicating that lifetime earnings inequality is increasing. Although the increases measured by the parametric approach are larger, 79% compared to 35% [see Table 1], the inequality measures are dierent, they cover dierent time periods, and they cover dierent lengths of earnings history. The primary benet of the parametric method is that a clearer picture of the change in the trends is available: the increase in lifetime earnings inequality appears to have occurred almost exclusively during the rst half of the 198's. Does equation (4), a measure of predicted lifetime inequality that can change from year to year, provide a useful measure of changes in actual lifetime inequality? From examining Figure 5, the answer is a qualied yes. Figure 5 depicts a world in which predicted lifetime inequality remained approximately level (around.15) for the period but then increased to a new level (around.275) for the period after 1985; the gure indicates that a regime shift may have occurred during the early 198's which increased lifetime earnings inequality. If predicted lifetime inequality were to remain at the new, higher level for a sucient period of time, then one should expect that the measured changes in predicted lifetime inequality will be reected in the actual lifetime earnings of individuals. However, such a conclusion should be quickly qualied because the predicted lifetime inequality measure is not intended to project what will occur outside of the sample period. The lifetime inequality results are also consistent with results from related research. Gittleman and Joyce (1996), using the 2-year matched CPS, nd that long run inequality (for ve year periods) increased gradually between and , and then increased rapidly through Buchinsky and Hunt (1996), examining annual inequality and mobility in NLSY, conclude that long term mobility declined from Gottschalk and Mott (1994), also using the PSID, nd that long run inequality increased (for a nine year period) when comparing the periods and All of these results are consistent with my conclusion that lifetime inequality is markedly higher during the latter half of the 198's. Mott and Gottschalk (1995) nd that a permanent component of earnings variance is higher in the 198's then in the 197's, but they nd that the permanent component declines substantially from 1982 to However, Mott and Gottschalk (1995) focus on inequality within education (and age) cells where I look at inequality overall; the results presented here could be consistent with Mott and Gottschalk (1995) if the returns to education increased suciently during the 198's, a possibility which is supported by other research [see Levy and Murnane (1992)]. 29 For comparability to the gure quoted for experience-adjusted inequality, the 79% is calculated as the increase from the average of the rst three years to the average of the last three years. If I compare just the rst year and the last year, I calculate an increase of 11% : instab.tex 17