Hours Worked and the U.S. Distribution of Real Annual Earnings 1976-2016
Iván Fernández-Val, Franco Peracchi, Aico van Vuuren, Francis Vella
aa r X i v : . [ ec on . E M ] F e b Hours worked and the U.S. distribution of real annualearnings 1976–2016
Ivan Fern´andez-Val ∗ Franco Peracchi † Aico van Vuuren ‡ Francis Vella § February 27, 2020
Abstract
We examine the impact of annual hours worked on annual earnings by decomposingchanges in the real annual earnings distribution into composition, structural and hourseffects. We do so via a nonseparable simultaneous model of hours, wages and earn-ings. We provide identification results and estimators of the objects required for thedecompositions. Using the Current Population Survey for the survey years 1976–2016,we find that changes in the level of annual hours of work are important in explainingmovements in inequality in female annual earnings. This captures the substantialchanges in their employment behavior over this period. The impact of hours on males’earnings inequality operates only through the lower part of the earnings distributionand reflects the sensitivity of these workers’ annual hours of work to cyclical factors. ∗ Boston University. † Georgetown University ‡ Gothenburg University § Georgetown University Introduction
Empirical studies of the hourly wage distribution in the U.S. assign the divergent patternsfor males and females at different points of the distribution to a variety of factors (see, forexample, Katz and Murphy 1992, Murphy and Welch 1992, Juhn, Murphy, and Pierce 1993,Welch 2000, Autor, Katz, and Kearney 2008, Acemoglu and Autor 2011, Autor, Manning,and Smith 2016, Murphy and Topel 2016, and Fern´andez-Val et al. 2019). These includeskill-biased technical change, international trade and globalization, the decline of the U.S.manufacturing sector, cohort size effects, the slowdown of the trend towards higher educa-tional attainment, the reductions in unionization rates and the associated union premium,and the falling real value of the minimum wage. Gottschalk and Danziger (2005) argue thathourly wages are most frequently examined in studies of inequality because they most accu-rately measure changes in the relative demand and supply of skills. However, an analysis ofhourly wages does not fully describe income inequality, which is perhaps more important forpolicy purposes. For example, understanding changes in poverty requires an examinationof family earnings, while evaluating growth at the top of the income distribution (as in, forexample, Piketty and Saez 2003) requires accounting for capital income. We extend theanalysis of inequality beyond hourly wages by examining changes in annual labor earningsin the U.S. over the period 1976 to 2016. While earnings are only part of total income, theyare nevertheless an important component. The Bureau of Labor Statistics reports the laborshare in total income to be 61.5 percent at the beginning of 1976 and 58.4 percent at theend of 2016.The differences between the changes over time in the respective distributions of hourlywages and annual earnings reflect movements in the distribution of annual hours of work.2hese can be potentially important when there are changes in the employment behavioron either the extensive or the intensive margins. During the 1976–2016 period we examinehere, the U.S. experienced a strong upward trend in female employment, large short-termswings in employment reflecting business cycle fluctuations, and the effects from the BabyBoomers entering the labor force, progressively aging and then exiting into retirement. Eachof these factors has had an important influence on the distribution of annual hours of work.We examine how they, combined with movements in the wage distribution, changed thedistribution of annual earnings.A challenge in evaluating the impact of annual hours worked on annual earnings is ac-counting for the selection bias resulting from only observing wages for those reporting positivehours of work. Fern´andez-Val, Peracchi, Van Vuuren, and Vella (2019), hereafter FPVV, ad-dress this in an investigation of changes in the real hourly wage distribution. Their paperemploys the methodology of Fern´andez-Val, Van Vuuren, and Vella (2019), hereafter FVV,to estimate the determinants of changes in the real wage distribution while accounting forthe selection associated with annual hours worked. It decomposes changes at different quan-tiles of the gender specific hourly wage distributions into three components: i) a “selectioneffect” that captures changes from the selection rule given the distribution of observed andunobserved individual characteristics, ii) a “composition effect” that reflects changes due tothe movements in the distribution of the observed and unobserved characteristics, and iii)a “structural effect” that measures changes in the conditional distribution of hourly wagesgiven changes in observed and unobserved individual characteristics. FPVV find that com-position effects have increased real hourly wages at all quantiles but that the patterns ofwage movements are primarily determined by the structural effects. Moreover, selectioneffects only appear at the lower quantiles of the female wage distribution. These various3omponents combine to produce a substantial increase in wage inequality.The present paper focuses on the role of annual hours in determining annual earnings. Wefollow FPVV and first model an individual’s annual hours of work. This provides estimatesof a control function which accounts for the selection bias when estimating the determinantsof hourly wages. The estimated model of annual hours can also be used to construct counter-factual annual hours. Using these estimated hours and wage equations, and exploiting thatannual earnings are the product of annual hours and hourly wages, we model annual earnings.We employ a nonseparable sample selection model with a censored selection variable basedon the approach of Imbens and Newey (2009) and FVV. We decompose the changes in realannual earnings into four components via the use of counterfactual earnings distributions.The first two, namely the structural and composition effects noted above, are those typicallyidentified in decomposition exercises, although their interpretation is more complicated ina multiple equation setting. We also provide two hours effects. The first is the extensiveeffect and captures the role of the work decision on annual earnings and measures how themovement from non employment to positive annual hours worked affects the earnings dis-tribution. This is called the selection effect in FPVV as it captures the impact of enteringor exiting employment. The second is the intensive effect and this captures how changes inthe annual hours of those working affects annual earnings.We model annual earnings using data from the Annual Social and Economic Supplementof the Current Population Survey, or March CPS, for the survey years 1976–2016. Thedata are a sequence of annual cross-sections containing information on earnings and hours ofwork in the previous year. FPVV find important structural and composition effects whenevaluating changes in the distribution of real hourly wages for these data. Our measure of In what follows we refer to the survey year rather than the year for which the data are collected. L -index. They decompose this measure into three components reflectingthe variance of wages, the variance of hours, and the covariance between hours and wages.Employing micro-level data for France, Germany, the U.K., and the U.S. for the period 1989–2012, they find differences in the dispersion of hours and wages across the four countries,different role and trends for the covariance between hours and wages, but similar patterns forthe effects of gender and education. Compared to the other three countries the level of hoursdispersion in the U.S. is moderate and it plays a smaller role in explaining earnings dispersionamong those with positive earnings. The relative contributions of wages and hours, and thecovariance between hours and wages, are relatively stable over time. Gottschalk and Danziger(2005) found similar results for the U.S. over the period 1975–2002. They focus on individualswith positive annual hours and earnings, ignoring the role of selection into employment. Weconsider these previous studies more descriptive as they ignore the endogeneity of working5ours. Also, in contrast to these studies we provide decompositions at different points of theearnings distribution.The next section discusses the CPS data and its features related to the annual earningsdistribution. Section 3 discusses issues related to the patterns of variation in annual hoursworked. Section 4 presents the model and discusses identification of the local objects ofinterest and the decomposition methodology. Section 5 presents the estimation approach.Section 6 reports and discusses the results. Section 7 concludes. We employ data from the Annual Social and Economic Supplement of the Current PopulationSurvey (or March CPS) for the 41 survey years from 1976 to 2016, which report annualearnings, weeks worked, and usual hours of work per week in the previous calendar year. Werestrict our analysis to those aged 24–65 years in the survey year. This produces a sample of1,794,466 males and 1,946,957 females, with an average annual sample size of 43,767 malesand 47,487 females. Annual sample sizes range from a minimum of 30,767 males and 33,924females in 1976 to a maximum of 55,039 males and 59,622 females in 2001.Annual hours worked last year (hereafter referred to as annual hours) are the product ofweeks worked and usual hours of work per week. This definition allows the same number ofannual hours to be produced by different combinations of hours and weeks. For example,we do not distinguish between an individual working 40 hours a week for 25 weeks fromone working 20 hours a week for 50 weeks. While FPVV provide evidence that this isunimportant for evaluating hourly wage rates in the U.S., in that hours do not appear tohave a direct influence on wage rates, it is important in evaluating the role of movements in6ours due to changes along the extensive and intensive margins. It is useful to highlight thedistinction between the conventional definition of unemployment in the monthly CPS andour employment measure based on annual hours of work last year. According to the former,people are classified as unemployed if: (i) they are not employed during the survey referenceweek, (ii) they are available for work during the survey reference week (except for temporaryillness), and (iii) either they made at least one specific, active effort to find a people jobduring the 4-week period ending with the survey reference week or were temporarily laidoff and expecting to be recalled to their job. This definition of unemployment does notdepend on the number of hours or weeks worked in the year before the survey, as recordedin the March CPS. Our employment measure means that an individual who works evenone hour for one week of the year while unemployed for the remaining weeks of the year isconsidered as employed. Thus, if the same individual worked 40 hours per week for 50 weeksin the next year, the resulting change in earnings would be assigned to the intensive hourmargin (assuming no further changes due to structural and composition effects). There isno extensive hour effect from increasing annual hours from one hour to 2000 hours a year.This is an important issue and we discuss it below.Individuals reporting zero hours worked last year generally respond that they were notin the labor force in the week of the March survey. As hourly wages are defined as the ratioof reported annual earnings and annual hours worked in the year before the survey they areonly available for those in the labor force. This has implications for those in the ArmedForces, the self-employed, and unpaid family workers as their annual earnings and annual FPVV investigate the robustness of their results to the construction of the annual hours variable byreplicating the analysis after restricting their sample to full-time full-year workers. This excludes observationsthat work many hours for one week or few hours for many weeks. They find that their results are robust tothe exclusion of these observations. A small fraction of individuals reporting zero annual hours of work last year also report being unemployedin the March survey. Most report that they were out of the labor force. A nnu a l e a r n i n g s ( × $ ) Males Females1980 1990 2000 2010020406080100120 Survey year A nnu a l e a r n i n g s ( × $ ) Males 1980 1990 2000 2010Survey yearFemalesD1 Q1 Q2 Q3 D9Earners sampleFull sampleFigure 1: Time profile of selected quantiles of annual earnings.9ercent at the top quartile and 40 percent at the top decile. The figures for males are similarto those in FPVV, suggesting the patterns are primarily driven by changes in hourly wagesand/or the changes in annual hours are working in the same direction as those generatingthe changes in hourly wages. While the increases at the higher quantiles are remarkable, thechanges at the lower quantiles are concerning. Over the period we consider, annual earningsdecrease by 10 percent at the lower decile, 17 percent at the lower quartile, and 7 percent atthe median.The most notable difference between these patterns in annual earnings and those reportedfor hourly wages by FPVV is that at lower quantiles for females there is an earnings increasenot reflected in wages. This suggests that changes in annual hours are more important indetermining earnings changes at lower quantiles. Figure 2 presents the distribution of annualhours for our sample period. The level of males’ hours are stable at all points of the hoursdistribution although this may partially reflect the shortcomings of using quantile methodsto describe data in the presence of bunching. However, they display a cyclical behavior whichis particularly important at lower quantiles. To relate this cyclical behavior to the businesscycle the recessions are marked with a shaded vertical bar. There has been a consistentincrease in female annual hours at all quantiles, except the median, with the largest increasesat the bottom of the distribution. Cyclical effects for females are weaker and are manifestedas occasional dips in otherwise upward trends. The median for males and the upper quartilefor females show no variation over our sample and this also partially reflects the bunching ofhours. Approximately 40 percent of males and 30 percent of females were working exactly2080 hours per year in 1976. This corresponds to 52 weeks of a 40-hour work week. Almost During the period we consider, the NBER’s Business Cycle Dating Committee distinguishes five reces-sions: January to July 1980 (6 months), July 1981 to November 1982 (16 months), July 1990 to March 1991(8 months), March to November 2001 (8 months), and December 2007 to June 2009 (18 months), known asthe Great Recession. A nnu a l h o u r s Males Females1980 1990 2000 2010050010001500200025003000 Survey year A nnu a l h o u r s Males 1980 1990 2000 2010Survey yearFemalesD1 Q1 Q2 Q3 D9Earners sampleFull sampleFigure 2: Time profile of selected quantiles of annual hours of work.11980 1990 2000 2010050010001500200025003000 Survey year A nnu a l h o u r s Males 1980 1990 2000 2010Survey yearFemalesQ1 Q2 Q3Figure 3: Time profile of mean annual hours of work by location in the annual earningsdistribution (e.g., Q1 is the mean of annual hours for individuals with annual earnings up tothe lower quartile, etc.).50 percent of males and 40 percent of females have annual working hours of exactly 2080by the year 2016. This increasing bunching of annual hours is an important feature of thedata that has recently received some attention in the macroeconomic literature (see Bick,Blandin and Rogerson 2019).Figure 3 presents mean hours for those in different quartiles of the earnings distribution.Thus, Q1 denotes mean hours for workers with labor income less than the lower quartile ofearnings, Q2 denotes mean hours for those with labor earnings between the first quartile andthe median, and Q3 denotes mean hours for those with labor earnings between the medianand the upper quartile. Male hours have not substantially increased at any point in theearnings distribution and the cyclicality of male hours is more evident for those with lower12980 1990 2000 20100 . . . . . .
91 Survey year E m p l o y m e n tr a t e Males FemalesFigure 4: Time profile of the fraction of people with positive annual hours of work.earnings. There are small increases at higher female earnings and larger increases at themedian and below.The previous figures reflect movements on the intensive margin of hours worked. Figure 4focusses on the extensive margin and plots the time profile of employment rates, defined asthe fraction of people working positive annual hours last year. For both males and females,the trends change around year the 2000. For males it goes from being relatively flat tonegative, while for females the change is from positive to negative.
To investigate the patterns of variation in annual hours, Figure 5 presents the standarddeviations of usual weekly hours of work and weeks worked, σ H W and σ W , along with theircorrelation, ρ H W ,W . The left-hand scale is for the standard deviations and the right-hand13980 1990 2000 20101618202224 Survey year S t d . d e v i a t i o n , c o rr e l a t i o n Males 1980 1990 2000 2010Survey yearFemales1980 1990 2000 2010Survey yearMales 1980 1990 2000 2010 00 . . . . σ H W σ W ρ H W ,W Figure 5: Standard deviation of weekly hours of work, σ H W , standard deviation of weeksworked, σ W , and correlation between weekly hours of work and weeks worked, ρ H W ,W .scale is for the correlation.For males there is slightly greater variability in weeks than in weekly hours, but bothshow clear evidence of strong cyclical movements around an upward trend. There is alsoa steady increase in the correlation between weekly hours and weeks, indicating that thesetwo components of annual hours are more closely tied at the end of the period than thebeginning. For females the standard deviations of both weekly hours and weeks are higherthan for males. In contrast to males, they show little cyclical movement and their behavioris dominated by trends. Interestingly, these trends are negative until about year 2000 atwhich point they become positive. Despite the large differences by gender in the variabilityof weekly hours and weeks, the behavior of their correlation is similar, namely steadilyincreasing over the period. 14980 1990 2000 201000 . . . .
81 Survey year V a r i a n ce , C o v a r i a n ce Males 1980 1990 2000 2010Survey yearFemales σ H σ H W σ W σ ln H W , ln W Figure 6: Variance of log annual hours of work, σ H , variance of log weekly hours of work, σ H W , variance of log weeks worked, σ W , and covariance of log weekly hours and weeksworked, σ ln H W , ln W .Figure 6 decomposes the variance of log annual hours into three components reflecting,respectively, the variance of log weekly hours, σ H W , the variance of log weeks, σ W , andthe covariance between log weekly hours and log weeks, σ ln H W , ln W . This evidence supportsthat from Figure 5. That is, there is greater variability for females than for males. For malesall three components display strong cyclical fluctuations but no trend, while for females theyall display a downward trend but no cyclicality. This suggests that male behavior is moreinfluenced by cyclical factors, while female behavior is more influenced by long-run trends.For both genders the primary source of the variation in log annual hours is σ ln H W , ln W ,followed by σ H W and σ W . The time trends in the two variance components differ bygender. They are flat or slightly positive for males while being strongly negative for females.15s a result of these trends the relative contributions of the three component show littledifference by gender by the end of the sample period. Our goal is to decompose changes in the real annual earnings distribution into contributionsreflecting the changing impact of labor force composition in terms of observable and unob-servable characteristics, structural effects reflecting these characteristics’ prices, and hourseffects capturing the level of annual hours worked. We first estimate a model of annual hoursfor each cross section. This provides the control function required to account for selectionin estimating the determinants of hourly wages over the sample of workers. We now outlineour model, introduce the control function and the other objects of interest, and describe thedecomposition of changes in the distribution of annual earnings.
Our model consists of the following three equations: Y = { H > } W H, (1) W = g ( X, E ) , if H > , (2) H = max { k ( X, Z, U ) , } , (3)where Y , W , and H denote annual earnings, hourly wages, and annual hours of work re-spectively, X and Z are vectors of observable explanatory variables, and { A } denotes theindicator function of the event A . The functions g and k are smooth but unknown, while E and U are potentially mutually dependent unobservable random vectors with stochastic16roperties described below. Equations (1)-(3) may be regarded as the reduced form of amodel of individual labor supply where the annual hours worked and the hourly wage rateare jointly determined as functions of the observable variables in X and Z , and the unob-servable variables E and U . We estimate functionals related to g and k noting that W isonly observed when H is positive.We make the following assumptions: Assumption 1 (a) ( X, Z ) are independent of ( E, U ) .(b) The function u k ( · , · , u ) is strictly increasing on the support of U .(c) U has a uniform distribution on [0 , . These assumptions are standard in this literature (see, for example, Imbens and Newey2009 and FVV). Assumption 1(a) restricts the endogeneity in equation (2) to operate via theselection
H >
0. The model could be extended to include additional endogenous explanatoryvariables in equation (2). Assumption 1(b) requires that H is continuous and, together withAssumption 1(a), restricts the dimension of the unobservables responsible for the endogeneityin equation (3). Given Assumption 1(b), Assumption 1(c) is an innocuous normalization(Matzkin, 2003).The control function for those who work positive hours, H >
0, is the random variable F H | X,Z ( H | X, Z ) where F H | X,Z ( h | x, z ) is the distribution of H conditional on X = x and Z = z evaluated at H = h . Formally: Lemma 1 (Control function)
The random variable V := F H | X,Z ( H | X, Z ) is a controlfunction for E conditional on H > . That is, E is independent of X and Z conditional on V and H > . U and V are identical when H >
0. Then, conditioning on X , Z and V , or equivalently on U , makes the statementthat k ( X, Z, U ) >
0, and hence
H >
0, deterministic. It implies that the distribution of E conditional on X , Z , V , and H > E conditional on X , Z and U . The assumption that E and U are independent from X and Z concludes the proof. The decomposition of the changes in the earnings distribution requires some structural func-tions from the model in (1)-(3). We first establish the region where these functions areidentified. We then relate the structural functions and distributions of the explanatoryvariables and control function to the distribution of earnings exploiting that Y = W H .We denote the support of a random variable in the entire population by calligraphicletters. Hence, X is the support of X . Also, X Z and
X V denote the respective jointsupport of the random vectors (
X, Z ) and (
X, V ). The support of the random vector Z conditional on X = x and V = v is denoted by Z ( x,v ) .We start with the following definitions: Definition 2 (Identification set)
Let
X V ∗ := (cid:8) ( x, v ) ∈ X V | ∃ z ∈ Z ( x,v ) : k ( x, z, v ) > (cid:9) . The set
X V ∗ represents the support of the random vector ( X, V ) for those individualswith a positive value of H . The size of this set depends on the availability of variables in Z that satisfy the exclusion restriction in equation (2) and their relevance in equation (3). Definition 3 (Local objects)
The local average structural function (LASF) of hourly wages s defined as: µ ( x, v ) = E [ g ( x, E ) | V = v ] provided that the expectation exists. The local distribution structural function (LDSF) ofhourly wages is defined as: G ( w, x, v ) = E [ { g ( x, E ) ≤ w } | V = v ] . Let W ( x ) = g ( x, E ), be the potential hourly wage when X = x . The LASF is the meanof W ( x ) conditional on V = v . It represents the mean wage that would be observed wheneveryone in the sample who has a value of V equal to v is assigned a value of X equal to x .The LDSF is the corresponding distribution function at w . Identification of the LASF andthe LDSF is established in FVV and is summarized by the following lemma. Lemma 4 (Identification of the local objects)
The LASF and the LDSF are identifiedfor all ( x, v ) ∈ X V ∗ . The proof of Lemma 4 relies on the result that the expectation defining the LDSF canbe made conditional on (
X, Z, V ) = ( x, z, v ) by Lemma 1. Then, for any ( x, v ) ∈ X V ∗ ,there is a z ( x, v ) ∈ Z x,v such that k ( x, z ( x, v ) , v ) >
0, so that the distribution of g ( x, E )conditional on V = v , i.e. the LDSF, is the same as the distribution of Y conditional on X = x , Z = z ( x, v ) and H > To quantify the role of hours in determining annual earnings we use the local objects tocharacterize counterfactual distributions and means of annual earnings that we would observe19f we change the distributions of the explanatory variables, control function, wages and hoursworked. We focus on distributions as means can be obtained by replacing the LDSF withthe LASF in the analysis that follows.We first rewrite the observed marginal distribution of earnings as a mapping of thedistributions of the explanatory variables, control function, wages and hours worked. By thelaw of iterated expectations: F Y ( y ) = P [ Y ≤ y ] = Z P [ Y ≤ y | X = x, Z = z, V = v ] dF X,Z,V ( x, z, v ) . Using Y = W H and H = k ( X, Z, V ) when k ( X, Z, V ) >
0, the definition of W from (2)gives: F Y ( y ) = Z k ( x,z,v ) > P (cid:20) g ( X, E ) ≤ yk ( X, Z, V ) | X = x, Z = z, V = v (cid:21) dF X,Z,V ( x, z, v )+ Z k ( x,z,v ) ≤ dF X,Z,V ( x, z, v ) . Then, by the definition of the LDSF G ( w, x, v ) : F Y ( y ) = Z k ( x,z,v ) > G (cid:18) yk ( x, z, v ) , x, v (cid:19) dF X,Z,V ( x, z, v ) + Z k ( x,z,v ) ≤ dF X,Z,V ( x, z, v ) . We define the counterfactual distributions of earnings as: G h q,r,s,p i ( y ) = Z k q ( x,z,v ) > G p (cid:18) yk r ( x, z, v ) , x, v (cid:19) dF sX,Z,V ( x, z, v ) ++ Z k q ( x,z,v ) ≤ dF sX,Z,V ( x, z, v ) , (4)which correspond to the distribution that we would observe if the labor force participation20ere as in year q , distribution of working hours as in year r , distribution of explanatoryvariables and control function as in year s , and distribution of wages as in year p . Forexample, the observed distribution of earnings in year t corresponds to G h t,t,t,t i = F tY . In theprevious expressions we use superscripts in the functions to denote the year from which thefunction is taken. For example, F sX,Z,V denotes the distribution of ( X, Z, V ) in year s .The identification conditions for these counterfactual distributions are based on threeseparate requirements. First, it must be possible to evaluate G p ( · , x, v ) for all ( x, v ) ∈ X V ∗ ,s .This requires that X V ∗ ,s ⊆ X V ∗ ,p . However, we only need to consider the elements of X V ∗ ,s that satisfy k q ( x, z, v ) >
0. Thus, the first restriction is (
X V ∗ ,q ∩ X V ∗ ,s ) ⊆ X V ∗ ,p . Second, itmust be possible to evaluate the function k r ( x, z, v ) for all the values of ( x, z, v ) in the firstintegral. By Assumption 1(b), k r ( x, z, v ) is identified for all ( x, z ) ∈ X Z r because we canset k r ( x, z, v ) = 0 for all v such that k r ( x, z, v ) <
0. Then, a similar reasoning as for the firstrestriction gives the second restriction (
X Z q ∩ X Z s ) ⊆ X Z r . Third, it must be possible toevaluate k q ( x, z, v ) > x, z, v ) ∈ X ZV s . By Assumption 1(b), this gives the thirdrestriction X Z s ⊆ X Z q . Finally, combining the three restrictions yields the identificationconditions ( X V ∗ ,q ∩ X V ∗ ,s ) ⊆ X V ∗ ,p and X Z s ⊆ ( X Z q ∪ X Z r ). Let Q h q,r,s,p i ( τ ) denote the counterfactual τ -quantile of earnings corresponding to the left-inverse of the counterfactual distribution y G h q,r,s,p i ( y ). The definitions above allow thefollowing decomposition of the observed change in the τ -quantile of earnings between the That is, Q h q,r,s,p i ( τ ) = inf { y ∈ R : G h q,r,s,p i ( y ) ≥ τ } , for τ ∈ (0 , t : Q h t,t,t,t i ( τ ) − Q h , , , i ( τ ) = [ Q h t,t,t,t i ( τ ) − Q h t,t,t, i ( τ )] | {z } [1] + [ Q h t,t,t, i ( τ ) − Q h t,t, , i ( τ )] | {z } [2] + [ Q h t,t, , i ( τ ) − Q h t, , , i ( τ )] | {z } [3] + [ Q h t, , , i ( τ ) − Q h , , , i ( τ )] | {z } [4] . (5)Term [1] in (5) is a “structural effect” reflecting earnings changes from movements in thedistribution of hourly wages for those who work conditional on the explanatory variables andcontrol function; term [2] is a “composition effect” reflecting changes in the distribution ofearnings resulting from changes in the distribution of the explanatory variables and controlfunction for those who work; term [3] is an “intensive hours effect” reflecting changes inearnings resulting from changes in the distribution of hours worked for those who workconditional on the explanatory variables and control function; while term [4] is an “extensivehours effect” capturing how individuals who change their employment status, from nonworking to working, increase their earnings from zero to a positive amount. FPVV discussthe role of composition, structural, and selection effects for the changes in the distribution ofhourly wages and highlight the complications which arise in a nonseparable model. These areeven greater here, as changes operating through hours directly enter the earnings equations.Accordingly, we further discuss their respective interpretations.The extensive hours effect corresponds to the selection effect in FPVV. It captures thedifference between the quantiles of earnings in year t and the quantiles of earnings that wouldhave prevailed if individuals from year t participated as in year 0 (based on their observed The extensive margin effect is positive as, given the choice of base years, it reflects the impact fromgoing from zero hours to positive hours worked on annual earnings. In FPVV the selection effect is theimpact on the observed wage distribution as we expand the number of people working. Thus in that contextthe effect can be negative. G h , , , i , thiseffect is always positive. That is, someone with a positive probability to be working in thebase year 0 should also have a positive probability in year t . By Assumption 1(c), those whodo not work in year 0, and therefore have a low level of the control function V , will have alow number of working hours in year t . A violation of this assumption would result in anunderestimation of the extensive hours effect.The intensive hours effect is the difference between the quantiles of earnings that wouldresult from changing the distribution of hours from year t to year 0, when individuals par-ticipate as in year t . More specifically, it changes the function k from year t to year 0,while keeping the functions G and F X,Z,V as in year 0 and the condition k ( x, z, v ) > t . As we move from the base year to year t this will capture differences in the hoursdistribution of those working. We highlight again that since we are employing annual hours,these differences may reflect either changes in weekly hours or unemployment spells.The composition effect contains the impact on earnings of changes in the distribution ofboth observable and unobservable characteristics that affect wages and/or working hours.For example, increases in the level of female higher education could shift the distribution ofearnings to the right due to both increases in the hourly wages and the number of workinghours. Due to the nonseparable and nonparametric nature of the model, it does not seempossible to disentangle these two effects. However, note that the higher educational levelswhich increased hours, and thus higher annual earnings, are interpreted here as compositioneffects and not hours effects.Finally, consider the structural effect. This captures the manner in which the individual’scharacteristics, both the explanatory variables and control function, map into the distributionof hourly wages and thus annual earnings. In separable models the manner in which the23ontrol function affects the distribution of wages is sometimes referred to as a componentof the selection effect. FPVV show that in nonseparable selection models it is not generallypossible to isolate the effect of the control function.It is important to clearly state what is, and what is not, included in the structuraleffects. When we consider changes in the hours distribution below, we are able to isolate thestructural and composition effects of the explanatory variables. When considering earnings,however, the structural effects only operate through their impact on wages. The impact of thestructural effects operating on earnings through hours are assigned either the interpretation ofextensive or intensive hours effects. Studies that estimate parametric models which accountfor selection sometimes attribute changes resulting from movements in the control functionand changes in its coefficient as selection effects. FPVV show that in a nonseparable modelthis attribution is generally not possible and argue that these selection effects should beinterpreted as structural effects. We follow FPVV and assign the changes in annual earningsoperating through the impact of the control function on wages to the structural effect. Weinterpret the impact of changes in the value of the control function as a composition effect.Consider the role of Assumption 1(c) in interpreting these various components in thesecross year comparisons. This assumption implies that individuals entering employment whengoing from a workforce with a low employment level to a workforce with a high employmentlevel, will always have lower hours than those who work in the “low employment” scenariolevel. If this assumption is violated, then we would underestimate the extensive hours effectsince we would underestimate the hours of those entering the labor market under the lessrestrictive selection rule. In contrast, the intensive hours effect will be overestimated sincewe will mistakenly consider an increase in the working hours of those that entered the labormarket as an increase of the working hours of those already in the labor market.24 Estimation
Our objective is to estimate the decomposition shown in (5). This requires estimation ofthe counterfactual distributions in equation (4) which are comprised of components fromdifferent samples. We start by rewriting this equation as: G h q,r,s,p i ( y )= Z (cid:16) F qH | X,Z (0 | x, z ) < v (cid:17) " G p yQ rH | X,Z ( v | x, z ) , x, v ! − dF sX,Z,V ( x, z, v ) + 1 , (6)where v Q rH | X,Z ( v | x, z ) is the inverse function of h F rH | X,Z ( h | x, z ). Next, we estimatethe components of this equation in the following manner: Step 1: Estimation of the control function.
We estimate the control function usinglogistic distribution regression (Foresi and Peracchi 1995, and Chernozhukov et al. 2013).That is, for every year s and every observation in the selected sample with H si >
0, we set: b V si = Λ( P s ⊤ i b π s ( H si )) , i = 1 , . . . , n s , where Λ( u ) = (1 + e − u ) − is the standard logistic distribution function, P si = p ( X si , Z si ) is a d p -dimensional vector of transformations of X si and Z si with good approximating propertiesand, for any h in the empirical support of H at time s , b π s ( h ) is the logistic distributionregression estimator: b π s ( h ) = arg max π ∈ R dp n s X i =1 (cid:2) { H si ≤ h } P s ⊤ i π + log Λ( − P s ⊤ i π ) (cid:3) . tep 2a: Estimation of the LDSF of hourly wages. Our estimation method is basedon a trimmed sample with respect to annual hours and we employ the following trimmingfunction: T si = { < H si ≤ h } , for some 0 < h < ∞ such that P [ T si = 1] >
0. The estimator of the LDSF of hourly wagesis: b G s ( w, x, v ) = Λ( m ( x, v ) ⊤ b θ s ( w )) , where m ( x, v ) is a d m -dimensional vector of transformations of ( x, v ) with good approximat-ing properties. The vector b θ s ( w ) is the logistic distribution regression estimator: b θ s ( w ) = arg max θ ∈ R dm n s X i =1 (cid:2) { W si ≤ w } M s ⊤ i θ + log Λ( − M s ⊤ i θ )) (cid:3) T si , and M si = m ( X si , b V si ). Step 2b: Estimation of the LASF of hourly wages.
The estimator of the LASF ofhourly wages is b µ s ( x, v ) = m ( x, v ) ⊤ b β s , where m ( x, v ) is a d m -dimensional vector of trans-formations of ( x, v ) with good approximating properties, b β s is the ordinary least squaresestimator: b β s = n s X i =1 T si M si M s ⊤ i ! − n s X i =1 T si M si W si , and M si = m ( X si , b V si ). Step 3: Estimation of the counterfactual distribution function of annual earnings.
Using the analogy principle, G h q,r,s,t i ( y ) is estimated by the empirical counterpart of (6),26amely: G h q,r,s,p i ( y )= 1 n s n s X i =1 (cid:16) Λ( P s ⊤ i b π q (0)) < b V si (cid:17) " b G p y b Q rH | X,Z ( b V si | X si , Z si ) , X si , Z si ! − + 1 , where b Q rH | X,Z ( v | X si , Z si ) = R ∞ { Λ( P s ⊤ i b π r ( h )) ≤ v } dh is the generalized inverse of h Λ( P s ⊤ i b π r ( h )). The standard errors and confidence intervals for these objects and the components ofthe decomposition can be calculated using the weighted bootstrap. This method obtainsthe bootstrap version of the estimator of interest by repeating all the estimation steps usingsampling weights drawn from a nonnegative distribution. More details regarding this proce-dure are provided in FVV. We do not provide standard errors for all of the decompositionsthat follow although we produce them for the decompositions of median earnings. We do soto obtain some indication of their magnitude.
We now report and discuss our empirical results. Section 6.1 decomposes changes over timein the quantiles of annual hours, Section 6.2 decomposes changes over time in the quantilesof annual earnings and Section 6.3 decomposes changes over time in a measure of earningsinequality. Section 6.4 discusses the main findings. The integral in the expression of b Q rH | X,Z ( v | X si , Z si ) is approximated by a Riemann sum at the samplevalues of H r . .1 Annual hours We estimate equation (3) separately by gender and survey year. The vector X includesthree dummy variables indicating the highest level of education (less than high school, somecollege, and college), a quadratic polynomial in age fully interacted with the educationdummies, a dummy variable indicating that the individual is not married or cohabiting, adummy variable indicating that the individual is nonwhite, and three dummy variables forthe region of residence (North-East, South, and Central). The vector Z includes the size ofthe household (in deviations from two), a dummy variable for the presence of own childrenin the household, a dummy variable for the presence of children under 5 years of age, and adummy variable for the presence of other household members besides spouse or dependentchildren. The reference case is a white high-school graduate, married or cohabiting, residentin a Western state, in a two-member household without children or other household members.The decomposition exercises require a base year for each gender. We follow FPVV andemploy 1976 for females and 2010 for males. Irrespective of the base year, we present thechanges relative to 1976. The female employment rate is at its lowest in 1976 and we assumethat females with a certain combination ( x, z, v ) of observed and unobserved characteristicsworking in this period will have a positive probability of working in subsequent years. We usethe same logic in choosing 2010, the bottom of the financial crisis, for males. As we employdifferent base years, we can compare trends but not hours and earnings across gender.We start with an examination of changes in mean annual hours of work. Figure 7 presentsthe relative changes in mean annual hours relative to 1976. For males they vary little over This decomposition only distinguishes between composition and structure effects as in DiNardo, Fortinand Lemieux (1996). The composition effect reflects changes in the distribution of annual hours resultingfrom changes in the distribution of the explanatory variables, while the structure effect reflects changesresulting from changes in the conditional distribution of annual hours given the explanatory variables. − . . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010 − . . . − . . . R e l a t i v e d i ff e r e n ce Males StructuralCompositionTotalFigure 8: Decomposition of the relative changes in the lower quartile of annual hours of workwith respect to 1976.period, as more than 25 percent of females work zero annual hours. Male hours are highlycyclical with a decrease of over 20 percent during the early 1980s and 40 percent duringthe Great Recession. The composition effect is positive, and most likely reflects increasingeducation attainments. However, the dominant component is the negative structural effectwhich follows the business cycle and somewhat resembles the surge in unemployment ratesduring recessions.Figure 9 shows the time profile of the relative changes in median annual hours for femaleswith respect to 1976. For males, which we do not present, the median remains constant at2080 annual hours. Although quantiles are less informative in characterizing distributionswhich are discrete or feature substantial bunching, changes over time in the quantiles canstill be indicative of the movements in the underlying distribution and this is why we employthem here. The female profile resembles that of the mean but the increase is much larger30980 1990 2000 2010 − . . . R e l a t i v e d i ff e r e n ce FemalesStructural Composition TotalFigure 9: Decomposition of the relative changes in median annual hours of work with respectto 1976.(over 140 percent) and reflects the large increase in both female employment rates and hoursworked by those working. The large contribution of the composition effect has implicationsfor earnings decompositions and we return to this below.Figures 10 and 11 provide the time profiles of the relative changes with respect to 1976 in,respectively, the upper quartile and the upper decile of annual hours of work. These figuresreveal two striking features common across gender. First, there is little variation at thesequantiles and there are long periods in which they remain unchanged. This reflects that theupper tail of the hours distribution has changed little over time. Given the magnitude of thechanges, both the structural and the composition effects play a negligible role. The absenceof these effects is important in understanding their impact on earnings.31980 1990 2000 2010 − . . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010Survey yearFemalesStructural Composition TotalFigure 10: Decomposition of the relative changes in the upper quartile of annual hours ofwork with respect to 1976. 32980 1990 2000 2010 − . . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010Survey yearFemalesStructural Composition TotalFigure 11: Decomposition of the relative changes in the upper decile of annual hours of workwith respect to 1976. 33 .2 Annual earnings
We now decompose the changes in the annual earnings distribution. We estimate the hourlywage equation (2) for the selected sample using the same conditioning variables in X asabove, with the variables in Z excluded for identification purposes. The estimated controlfunction b V and its square are included, in addition to their product with the other includedvariables. We employ the same base years as above. The earnings decompositions requireestimates from the annual hours equation and the selection-corrected hourly wage equation.The results for annual hours are above and those for hourly wages are provided in FPVV. The bunching at 2080 working hours per year probably reflects the convenience for anindividual to respond that he/she works 40 hours per week for 52 weeks per year. However,it creates a difficulty in predicting an individual’s working hours given his or her observedcharacteristics and control function. Accordingly, we modify our method as follows.Suppose H ∗ is the actual number of working hours of an individual, while H is theobserved number of working hours. Hence, H ∗ is the left-hand side variable as in (3).Furthermore, assume the relationship between actual and observed working hours is: H = j if h j < H ∗ ≤ h j +1 . From the monotonicity assumption it follows that h j < H ∗ ≤ h j +1 implies that the controlfunction V is between the values v j ( x, z ) and v j +1 ( x, z ). From the definition above, this holdswhen H = j . We also know that V is uniformly distributed between these values althoughwe are not able to estimate the control function of the individual observations. However, it FPVV illustrate the lack of sensitivity of the wage equation decompositions to various specificationissues. These included the choice of exclusion restrictions, changes to the sample, and the exclusion ofannual hours from the wage equation.
34s possible to estimate the values v j ( x, z ) and v j +1 ( x, z ) for every j based on the observedlevels of working hours. This suggests that we can randomly sample the values of V betweenthe two estimated values to obtain draws of the control function. We use these sampledvalues rather than the estimated values based on H to calculate the counterfactual earningsdistributions.Figure 12 presents the decomposition of the changes in mean annual earnings relative to1976. We retain the same vertical scale to allow easier comparisons. Mean male earningsare cyclical and highly variable, with increases and decreases of up to 10 percent. Overthe sample period they increase by 5 percent. The trend in mean male earnings is stronglycorrelated with the structural effect operating through wages. The composition effect ispositive with an inverse U-shaped profile. The absence of composition effects in the changesin mean male hours indicates that the positive composition effect in mean male earnings isalso operating through wages. There is no evidence of an extensive hours effect for meanmale earnings while the intensive hours effect is cyclical but small.In contrast to the modest growth for males, female mean earnings increase by over 70 per-cent. This reflects a steadily increasing composition effect which accounts for over half ofthis increase. Important contributions also come from the structural effect and the intensivehours effect. Surprisingly, the contribution from changes in the extensive margin are small,reflecting our earlier discussion of what the hours effects capture. The large compositioneffects are interesting and, as we saw above, they also reflect their impact on mean hours.Moreover, FPVV suggest that they are important for females at all points of the wage dis-tribution. Perhaps the most interesting feature of this figure is the large intensive hourseffect at the mean. It indicates that the shift in the hours distribution for those working hasincreased females earnings by 20 percent. 35980 1990 2000 2010 − . . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010 − . . . As the female employment rate is as low as 60 percent in some periods, we are unable toconduct our decomposition for females at the lower quartile.The figure provides a bleak portrait of the labor market experience for males at thelower quartile of annual earnings. There are several instances of large decreases in earningsresulting in a substantial reduction of around 70 percent from 1976 to 2010 before a smallrebound at the end of the period. The two most striking decreases occur at the beginningof the 1980s and during the Great Recession. The structural effect, operating through realwage rates decreases, is large. The composition effect is small and generally positive untilthe year 2009.Consider now the impact of hours. Since there are relatively small changes in maleparticipation rates, there is little impact along the extensive margin. Perhaps the mostinteresting feature is the impact of movements on the intensive margin, recalling that thisnot only captures variations in weekly hours but potentially also periods of zero hours perweek for some fraction of the year. This intensive hours effect is negative and frequentlylarge. More strikingly, the intensive margin accounts for the largest decrease in annualearnings at Q1, which occurs in 2010. The intensive hours effect is responsible for almosthalf of this notable decline. We stress again that this decline reflects both increased weeks Let Q . be the lower quartile of earnings for the full sample. This value corresponds to the p th quantilefor the earners sample, where p = F Y ( Q . | Y > F Y ( y | Y >
0) = ( F Y ( y ) − F Y (0)) / (1 − F Y (0))and F Y ( Q . ) = 0 .
25, it follows that p = (0 . − F Y (0)) / (1 − F Y (0)) < .
25. From Figure 4, F Y (0) rangesbetween 0.1 and 0.15, so we obtain values for p between 0.11 and 0.17. − . . . R e l a t i v e d i ff e r e n ce Males ExtensiveIntensiveStructuralCompositionTotalFigure 13: Decomposition of the relative changes in the lower quartile of annual earningswith respect to 1976.of zero work and reduced hours in working weeks.Figure 14 shows the decomposition of the changes in median annual earnings with respectto 1976. For males, they fall by approximately 20 percent over our period. Moreover thepatterns of change follow closely the business cycle. Interestingly, although unsurprisingly,the changes in male median earnings largely reflect the structural component. The structuralcomponent is more negative than the total effect as the composition effect is positive andgenerally increasing over this period. The composition effect is approximately 10 percent by2016 and most likely captures the impact of increasing education levels on wages. The profileof the intensive hours effect is largely negative and mirrors the business cycle. Moreover,there is no evidence of any effect on the extensive margin recalling that some fraction of thereported intensive hours effect reflects an increase in the number of zero weeks of positivehours. 38980 1990 2000 2010 − . . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010 − . . . − . . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010 − . . . − . . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010 − . . . − − . . R e l a t i v e d i ff e r e n ce Males 1980 1990 2000 2010 − − . . The evidence above documents the drastic changes in the distribution of annual hours ofwork and annual earnings. We now examine the implications of these changes for the levelof earnings inequality. Studies of inequality typically examine the ratio of quantiles aboveand below the median, and how they vary over time. Popular choices are the quartile ratio(Q75/Q25) or the decile ratio (D9/D1), although their selection is somewhat arbitrary. Giventhe inclusion of those working zero hours in our sample, the female participation rate of 56.5in 1976 does not allow for comparisons for females involving earnings below the fourth decile.Accordingly, we focus on relative changes with respect to 1976 in the ratio of the upper decileto the median for both males and females. These are provided in Figure 17.44he increase in male inequality of around 60 percent is substantial. Although all com-ponents are positive, this increase largely reflects the role of the structural component. Theevidence above, combined with that of FPVV, suggests this captures the role of the structuralcomponent in wage growth. That is, male wage rates at the upper quantiles have drasticallyincreased relative to those at lower quantiles. Unsurprisingly, there is little evidence of hoursand composition effects. The hours effects at the median and the 9th decile do not appearto be important and the composition effects appear to offset each other.The corresponding figure for females is strikingly different in several ways. First, there isa drastic decrease of over one hundred percent in this measure of inequality. The fall occursprimarily during the 1976 to 1990 period, although it is maintained for the remainder ofthe time examined. This large effect is the result of increased participation rates and hoursworked. Interestingly, this is captured either by the composition effects operating throughhours or the intensive hours effect directly. For different parts of the sample period thesetwo effects take on the dominant role. Note again that the intensive margin appears to becapturing those who were working relatively few hours early in the sample period but greatlyincreased their hours of work in subsequent years. Second, the structural effect is relativelyunimportant for this measure of female earnings inequality. The evidence in FPVV suggeststhe structural effects should be increasing inequality as those at the upper quantiles of thewage distribution have experienced much larger wage increases from this source than thoselower in the wage distribution. This, however, does not appear to be the case when weconsider the median and ninth deciles. Finally, the extensive hours effect has the expectedeffect but is small. This highlights the subtle distinction between the intensive and extensivemargins given our construction of the annual hours variable.45 .4 Discussion
The patterns of change in the distribution of annual earnings are very different for malesand females. This is especially so for the period 1976 to 2000. While the male distributionis dominated by cyclical fluctuations, the female distribution is driven by long-run trends.Another notable difference is the contribution of composition, structural, and hours effects.Structural effects closely reflect the business cycle and are more important for males, whilecomposition effects closely reflect long-run demographic and social trends and are moreimportant for females. Further, while for males structural and composition effects havedifferent signs, for females they are both positive and reinforce each other. For females, theintensive hours effect is important, especially until the early 2000s, but is smaller than thecomposition effect. However, recall that the increased education which affects the numberof hours worked is reflected here in the composition effect. For both genders, the extensivehours effect is negligible at all quantiles considered. While the pre-2000 trends differ greatlyby gender, the post-2000 trends appear more similar.The patterns of change in the distribution of annual earnings are also different acrossquantiles of the earnings distribution. Male annual earnings have been falling steadily atthe median and below, remained stable around the upper quartile, and increased at theupper decile and above. For females, they have been rising over the whole distribution.Female annual earnings have increased more at the bottom of the distribution than at thetop with the larger increases occurring in the first half of our period. Identifying the variouscomponents of this drastic increase in female earnings is arguably our most striking empiricalcontribution. 46
Conclusions
This paper examines the relationship between the changes in the distribution of annual hoursof work and those in the distribution of annual earnings. This is done through a decompo-sition of movements in the distribution of annual earnings into composition, structural andhours effects. This requires the estimation of a nonseparable model of hours and wages inthe presence of a censored selection rule. Our empirical work has a number of interestingresults. Changes in the hours of annual work are important for the lower quartile of maleannual earnings but their importance diminishes as we examine higher quantiles. At thelower quartile the effects operate through the intensive margin although the nature of theannual hours of work variable allows movements at the intensive margin to incorporate spellsof unemployment. While the upper decile to median ratio for male annual earnings increasesby over 50 percent for our sample period a very small fraction of this can be attributedto changes in the male hours distribution. This increase in annual earnings inequality ap-pears to be almost entirely to structural effects operating through the hourly wage equation.Changes in the female distribution of annual hours have increased earnings at all quantilesof the female annual earnings. These effects are generally operating through the intensivemargin. The female upper decile to median ratio decreases dramatically indicating a drasticreduction in earnings inequality. This reduction is due to large effects at the intensive marginof annual hours of work but also to composition effects, such as increased levels of education,which increase wages at all quantiles but also increase the level of annual hours of work.47 eferences [1]
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American Economic Review Papers and Proceedings , 444–449.50 ppendix A : Bootstrapped confidence intervals for de-compositions at median annual earnings Figure 18: 95 percent bootstrap confidence intervals at the median for females.1980 1990 2000 201000 .
51 Year R e l a t i v e d i ff e r e n ce Extensive 1980 1990 2000 2010YearIntensive1980 1990 2000 201000 .
51 Year R e l a t i v e d i ff e r e n ce Structural 1980 1990 2000 2010YearComposition51igure 19: 95 percent bootstrap confidence intervals at the median for males.1980 1990 2000 201000 .
51 Year R e l a t i v e d i ff e r e n ce Extensive 1980 1990 2000 2010YearIntensive1980 1990 2000 201000 .
51 Year R e l a t i v e d i ff e r e n cece