Measuring wage inequality under right censoring
MMeasuring wage inequality under right censoring ∗ Jo˜ao Nicolau a , Pedro Raposo b and Paulo M. M. Rodrigues ca ISEG-Universidade de Lisboa and CEMAPRE b Cat´olica Lisbon School of Business and Economics c Banco de Portugal and Nova School of Business and Economics
April 28, 2020
Abstract
In this paper we investigate potential changes which may have occurred over the last twodecades in the probability mass of the right tail of the wage distribution, through theanalysis of the corresponding tail index. In specific, a conditional tail index estimatoris introduced which explicitly allows for right tail censoring (top-coding), which is afeature of the widely used current population survey (CPS), as well as of other surveys.Ignoring the top-coding may lead to inconsistent estimates of the tail index and tounder or over statements of inequality and of its evolution over time. Thus, having a tailindex estimator that explicitly accounts for this sample characteristic is of importanceto better understand and compute the tail index dynamics in the censored right tailof the wage distribution. The contribution of this paper is threefold: i) we introducea conditional tail index estimator that explicitly handles the top-coding problem, andevaluate its finite sample performance and compare it with competing methods; ii)we highlight that the factor values used to adjust the top-coded wage have changedover time and depend on the characteristics of individuals, occupations and industries,and propose suitable values; and iii) we provide an in-depth empirical analysis of thedynamics of the US wage distribution’s right tail using the public-use CPS databasefrom 1992 to 2017.
Keywords : Wage inequality, tail index, top-coding, current population survey, wagedistribution, Pareto, occupations
JEL classification:
C18, C24, E24, J11, J31 ∗ We thank Daron Acemoglu and participants of the Workshop on
New Frontiers in Statistics of Ex-tremes and of the IX t Workshop in
Time Series Econometrics for helpful comments and suggestions, andFilipe Caires for excellent research assistance. This work was funded by Fundao para a Cincia e Tecnolo-gia through project number UID/Multi/00491/2019, PTDC/EGE-ECO/28924/2017, UID/ECO/00124/2013,UID/ECO/00124/2019, UID/GES/00407/2013 and Social Sciences DataLab (LISBOA-01-0145-FEDER-022209), POR Lisboa (LISBOA-01-0145-FEDER-007722, LISBOA-01-0145-FEDER-022209), and POR Norte(LISBOA-01-0145-FEDER-022209). a r X i v : . [ ec on . E M ] A p r Introduction
The sharp rise in overall wage inequality in the second half of the 20th century has become astylized fact (Autor (2019) and Goos et al. (2014)). Wage inequality growth in the 1980s wasfollowed by a slowdown in the 1990s as a result of divergent trends in the bottom and topof the wage distribution. Both the 90/50 and 50/10 indexes grew rapidly in the early 1980s,and although lower tail inequality virtually stopped growing after 1987 upper-tail inequalitykept rising. The deceleration in inequality growth observed in the 1990s resulted mainlyfrom polarization, i.e., from an abrupt stop or reversal of inequality growth in the lower-tailcoupled with a sustained secular rise of the upper-tail inequality. According to Autor et al.(2008) between 1963 and 2005 the 90th percentile wage rose by more than 55% relatively tothe 10th percentile for both men and women.Existing empirical evidence suggests that the rise in wage inequality is largely explainedby shifts in the supply and demand for skills (Goos and Manning (2007)), and by the erosionof labour market institutions (e.g. unions and minimum wage) (Kalleberg (2011)). It isdocumented that the increase in inequality in the 1980s was the result of a secular rise inthe demand for skill which faced an abrupt slowdown in the relative supply of high-skilledworkers (college or equivalent) in the form of lower attainment and of smaller labor-enteringcohorts which originated expanding wage differentials (Autor et al. (2008); Katz and Murphy(1992); Card and DiNardo (2002); and Acemoglu and Autor (2011)).The monotonic increase of inequality until the late 1980s followed by the divergent evolu-tion in the top and lower half of the distribution is robust to different measures and samples. Steady growth in the upper-tail inequality can also be seen from the rising share of wagespaid to the top 10% and 1% earners (Piketty and Saez (2003)). However, literature basedon public-use CPS data has produced a less than perfect picture of the right tail of the wagedistribution because of the top-coding (Armour et al. (2016)).The CPS wage data has historically been censored at the top (top-coded) and ignoringthis fact or not adequately handling it may result in inconsistent tail index estimates, lead tounderstatements of inequality and affect the estimates of its dynamics (Feng et al. (2006)). In addition, top-coding has changed over time. For instance, the top-coded wage was set at$1923 in 1997 and changed to $2884 from 1998 onward. But even during periods of constantnominal top-coding the data may hide changes in inequality (Levy and Murnane (1992)).While some authors have tried to address the top-coding issues by restricting the sam-ple under analysis, the method presented in this article makes use of the complete set ofinformation available from the public use CPS data, for every year, in a time-consistent fash- The result holds for male and female samples separately, considering weekly wages of full-time workersas well as for the March CPS samples (Autor et al. (2006)). Parker (1999) developed a model of wages in which wages follow a Generalized Beta Distribution of thesecond kind (GB2). Bordley et al. (1995) show that GB2 exhibits better fit to US wage data than alternativedistributions. Because the authors are modelling the distribution of total wage and not its components, theyonly need to know whether total wages are censored or not, and therefore do not need to be concerned withconsistency problems in categories as in Burkhauser et al. (2004).
To reduce the top-coding bias, researchers interested in measuring long-term trends in wagestypically impute top-code values to create a consistent series. Until recently one of fourapproaches has in general been adopted in the literature: (1) the top-coding problem isignored i.e., top-coded observations are dropped (see e.g. Jensen and Shore (2015)); (2) an ad hoc adjustment of the top-coded wages is made (e.g. Lemieux (2006) multiplied top-codedhourly wages by 1.4, and Autor et al. (2008) multiplied top-coded weekly wages by 1.5); (3)a Pareto distribution is used to estimate wages at the top of the distribution (e.g. Bernstein2nd Mishel (1997), Piketty and Saez (2003)); and (4) cell means or rank-proximity swappeddata based on the still-censored internal CPS data is used (e.g. Larrimore et al. (2008) andBurkhauser et al., 2008); for a discussion and shortcomings of these approaches see, interalia , Burkhauser et al. (2010) and Armour et al. (2016).In a recent contribution Armour et al. (2016) proposed an alternative approach whichconsists in estimating the tail index of a censored Pareto distribution. To briefly illustratethe procedure consider first the survival function, F , of a Pareto distribution F ( y ) := P (cid:0) Y i > y (cid:1) = (cid:18) y y i (cid:19) α , where y i ≥ y > α > f Y ( y ) = ( αy α ) / ( y α +1 i ) . A large number of tail in-dex estimation procedures is available in the literature. One widely used approach is theconditional maximum likelihood estimator (MLE) proposed by Hill (1975), (cid:98) α Hill := (cid:34) m m (cid:88) j =1 log y ( j ) − log y (0) (cid:35) − (2.2)where m is the number of largest order statistics used in the estimation of α , y ( j ) , j = 1 , ..., m are the largest m order statistics and y is the tail cut off point.However, recognizing the limitations of this approach when the data is top-coded, Armouret al. (2016) proposed an alternative method, which consists of an adaptation of the Hill esti-mator taking into consideration the censoring. This approach provides an unbiased estimateof the censored Pareto parameter, α , while using all available information. In specific, in thecase of a censored sample the outcome variable is, ω i = (cid:40) y i if y ≤ y i < y c y c if y i ≥ y c , (2.3)where y is the tail cut off point and y c the top-coded value. Hence, the density function ofthe censored Pareto distribution is, g Y ( ω i ) = (cid:18) αy α ω α +1 i (cid:19) I ( y ≤ ωi
0, where ˙ L ( y ; x ) = ∂ L ( y ; x ) /∂y. Y i conditional on X i is, f ( y | x ; θ ) = α ( x ) y − α ( x ) − L ( y ; x ) − y − α ( x ) ˙ L ( y ; x ) . (2.9)Considering (2.8) and assuming that y is sufficiently large, it follows that the density in (2.9)can be approximated as, f ( y | x ; θ ) ≈ c ( x ) α ( x ) y − α ( x ) − ; (2.10)see also Wang and Tsai (2009). Thus, the conditional probability function of Y i given X i and Y i > y can be approximated as, f ( y | x ; θ ) ≈ α ( x )( y/y ) − α ( x ) − , (2.11)where y is the threshold that controls the sample fraction used for estimation. Note that(2.11) is the approximate conditional Pareto density function of an unrestricted randomvariable and thus, its use when some form of censoring (such as right censoring in the CPSdatabase) is imposed on the data will originate inconsistent tail index parameter estimates.In the censored case, rather than observing the outcome y i , as in the previous section, weeffectively observe w i as defined in (2.3). In this context, the adequately adjusted conditionalPareto density function is, g ( ω i | x i , y c , θ ) := f ( ω i | x i , y c , θ ) I ( y ≤ w i Under Assumptions (A1) - (A2) it follows that n − / Σ − / y Λ − / ( (cid:98) θ − θ ) d → N ( , I p ) , where Λ := E ( e i | x i ) and e i = exp ( x (cid:48) i θ ) log (cid:16) w i y (cid:17) − , for I { y ≤ w i Under the same conditions of Theorem 2.1 as n → ∞ it follows that, T j = n − / (cid:16) Σ − / y ,jj (cid:17) Λ − / (cid:98) θ j d → N (0 , . (2.16) where Σ − / y ,jj corresponds to the ( j, j ) th element of the Σ − / y matrix. A further important and not immediately obvious aspect of the methodology just describedrelates to the computation of the partial effects of the covariates used in the conditional tailindex regression. In specific, for ease of presentation consider F ( y | x ; θ ) := P ( Y > y | x ; θ ) = (cid:18) y y i (cid:19) α ( x ) . x to be a scalarand continuous. Thus, to measure the impact of x on α ( x ) and subsequently on F ( y | x ; θ ),consider δ := ¯ F ( y | ∆ x + x ; θ ) − ¯ F ( y | x ; θ )¯ F ( y | x ; θ ) × 100 (2.17)where y is an extreme value, say the (1 − u ) quantile, with u ∈ (0 , y = (1 − u ) α ( x ) y . (2.18)Hence, δ in (2.17) measures the probability’s percentage variation of an extreme value dueto a variation of x , ∆ x . For example, considering u = 0 . 15 and ∆ x = 1 , if δ = 20% thenP( Y > y ) , where y is the 0.85 quantile, increases by 20% as a result of ∆ x = 1. Therefore,the variation of x increases the likelihood of observing extreme values by 20%.For computational purposes, assuming that α ( x ) := exp ( φ ( x )), and φ ( x ) is some functionof x , we show in the appendix that δ ( u ) = (cid:104) (1 − u ) φ (cid:48) ( x )∆ x − (cid:105) × . (2.19)For instance, in the multivariate case, φ ( x ) := x (cid:48) β , where x is a p × α ( x ) = exp ( x (cid:48) β ), and the impact of x j on α ( x ) is, δ j ( u ) = (cid:104) (1 − u ) β j ∆ x − (cid:105) × . (2.20)Thus, a negative (positive) coefficient increases (decreases) the likelihood of having moreextreme values, i.e. β j < β j > 0) implies δ j > δ j < 0) (this is also obvious from theimpact on α ( x ) since if α ( x ) decreases (increases), the right tail becomes (less) heavier). Remark 1. If we use only a portion of the sample to estimate the model, say for example,all observations larger than y , then y is the quantile of order 1 − u of the conditionaldistribution P ( Y < y | Y > y ) , i.e. P ( Y < y | Y > y ) = 1 − u. To determine the quantileorder of the unconditional distribution P ( Y < y ) we use the relation P ( Y < y | Y > y ) =1 − u ⇒ P ( Y < y ) = P ( Y < y ) + (1 − u ) P ( Y > y ) . In the empirical analysis below weuse all observations larger than the empirical quantile of 0.80 (see also Mishel et al. (2013)),so that P ( Y > y ) = 0 . 20 in the above formula. Hence, when u = 0 . 15 and u = 0 . , we areactually analysing the 96th and 97th quantile, respectively, of the unconditional distribution. ♦ Remark 2. When the covariate considered is discrete (e.g. a dummy variable) a simpleadaptation of (2.20) leads to the following formula, which measures the impact of group d = 1over d = 0, δ ( u ) := (cid:104) (1 − u ) α ( x ; d =1) α ( x ; d =0) − − (cid:105) × . Given that α ( x ; d = 1) and α ( x ; d = 0)depend on x , we also need to provide values for x . One possible solution is to replace x byits respective averages. ♦ Monte Carlo simulation In this section we evaluate the finite sample properties of the procedures and their perfor-mance in imputing mean wages above the top-code. To evaluate the performance of the conditional tail index estimator introduced in the previoussection, we conduct an in-depth Monte Carlo analysis using several data generation processes(DGPs). In specific, data is simulated from the general framework, y i ∼ D ( α ( x i )) (3.1) α ( x i ) = exp ( β + β x i ) , x i ∼ U (0 , 1) (3.2)where β = β = 1 and the k k ∈ (0 , D ( . ) closely follow a Pareto distribution. We consider the case of right censoring givenby the censoring threshold y c so that the sequence { y i } is not completely observed. Instead,we observe w i = min ( y i , y c ).To be more precise about the framework used to generate the data, we consider that D ( . ) in (3.1) is either a Pareto or a Burr distribution and generate samples of size n ∈{ , , , } . Moreover, we censor the sample considering y c = { ˆ q y . , ˆ q y . } which corresponds to the 95th and 99th empirical quantile of y . For estimation of the tailindex we use the (cid:98) kn (cid:99) largest observations, with k = 0 . k = { . , . , . } for the Burr. In the case of samples generated from aPareto distribution we could have set k = 1, however, a lower value was considered in orderto mimic the conditions typically found in empirical analysis.Based on the specifications described above we generated 10,000 sequences of { y i } and { w i } of size n and use in each iteration three estimation methods:i) The tail index regression of Wang and Tsai (2009) applied to the sequence of { y i } . Wedefine the resulting estimator as ˆ α . This method should provide the best results sinceit is applied to the original uncensored data.ii) The censored tail index regression introduced in this paper applied to the sequence { w i } . The resulting estimator is denoted as ˆ α c .iii) The tail index regression of Wang and Tsai (2009) applied to the censored data { w i } .The resulting tail index is defined as ˜ α . This approach will be useful in providinginformation on the impact of neglecting the censoring on the tail index estimates. The cumulative Burr distribution function considered in the simulations is F ( x ) := 1 − (1 + x − αρ ) ρ andthe corresponding probability density function f ( x ) := x − − αρ (1 + x − αρ ) − ρ α . β and β in (3.2)computed based on the three approaches described in i), ii) and iii). The first observationwe can make is that, in general, the largest bias and RMSEs (regardless of considering β or β ) result from the use of the approach described in iii), i.e., when the censoring is ignored.On the other hand, it is interesting to observe that the difference in the bias and RMSEsobtained from the approaches described in i) and ii) are relatively small, which suggest thatthe estimation approach which accounts for the censoring produces results close to thoseobtained when the sample without censoring is used for estimation as is the case in i).Moreover, this Table also shows that the bias remains relatively stable and does notdecrease as n increases. There are however different patterns according to the values of k and y c . For instance, in Cases 3 to 6, which use the Burr distribution as DGP, a small valueof k tends to improve the estimation results given that the tail of the Burr distribution getscloser to the tail of a Pareto distribution.To further evaluate the estimation performance of the three estimation approaches in i),ii) and iii), Figure 1 plots the ratios of the RMSEs of the α estimates obtained under thethese estimations approaches. In specific, the ratios are, Ratio RM SE ( ˆ α c ( x i )) RM SE ( ˆ α ( x i )) , Ratio RM SE ( ˜ α ( x i )) RM SE ( ˆ α ( x i )) . Since ˆ α , obtained as described in i), is the best estimator, Ratio Ratio n. However, Ratio 1 is just slightly above 1, which meansthat the censored estimator performs very well and mimics closely the behavior of the bestestimator, ˆ α , although the former is based on the censored data. On the contrary, Ratio 2 issubstantially higher than 1, which means that, ignoring the censoring when estimating thetail index produces an inconsistent estimator; see Figure 1.The censoring threshold y c also impacts the estimation results, i.e., the lower its value,the greater is the impact of censoring on estimation, and Ratio 2 tends to be larger (see, forexample, the results for Case 4 in Table 1 and Figure 1).9able 1: Bias and RMSE of estimators Case 1: DGP Pareto (k=0.20 and y c = Q y ( . ) ) ˆ α ( x i ) ˆ α c ( x i ) ˜ α ( x i ) ˆ α ( x i ) ˆ α c ( x i ) ˜ α ( x i ) ˆ α ( x i ) ˆ α c ( x i ) ˜ α ( x i ) ˆ α ( x i ) ˆ α c ( x i ) ˜ α ( x i )n 2500 5000 10000 50000bias( β ) 0 . . . . − . . . − . . . − . . β ) 0 . . − . . . − . . . − . . . − . β ) 0 . . . . . . . . . . . . β ) 0 . . . . . . . . . . . . Case 2: DGP Pareto (k=0.20 and y c = Q y ( . ) ) n 2500 5000 10000 50000bias( β ) 0 . − . . . − . . . − . . . − . . β ) 0 . . − . . . − . . . − . . . − . β ) 0 . . . . . . . . . . . . β ) 0 . . . . . . . . . . . . Case 3: DGP Burr ρ =-2 (k=0.05 and y c = Q y ( . ) ) n 2500 5000 10000 50000bias( β ) 0 . − . . − . − . . − . − . . − . − . . β ) 0 . . − . . . − . . . − . . . − . β ) 0 . . . . . . . . . . . . β ) 0 . . . . . . . . . . . . Case 4: DGP Burr ρ =-2 (k=0.10 and y c = Q y ( . ) ) n 2500 5000 10000 50000bias( β ) − . − . . − . − . . − . − . . − . − . . β ) 0 . . − . . . − . . . − . . . − . β ) 0 . . . . . . . . . . . . β ) 0 . . . . . . . . . . . . Case 5: DGP Burr ρ =-2 (k=0.20 and y c = Q y ( . ) ) n 2500 5000 10000 50000bias( β ) − . − . . − . − . . − . − . . − . − . . β ) 0 . . − . . . − . . . − . . . − . β ) 0 . . . . . . . . . . . . β ) 0 . . . . . . . . . . . . Case 6: DGP Burr ρ =-2 (k=0.20 and y c = Q y ( . ) ) n 2500 5000 10000 50000bias( β ) − . − . . − . − . . − . − . . − . − . . β ) 0 . . − . . . − . . . − . . . − . β ) 0 . . . . . . . . . . . . β ) 0 . . . . . . . . . . . . Note : ˆ α ( x i ) is the tail index regression estimate considering the complete sample of data (with no censoring); ˆ α c ( x i ) is the censored tail indexregression estimate; and ˜ α ( x i ) is the uncensored tail index regression estimate computed from censored data. n corresponds to the total sample size, k to the % of observations used for the tail index estimation and (cid:98) kn (cid:99) is the effective number of observations used in the estimation of the tail index. y c is the censoring value used and Q y ( τ ) corresponds to the τ th quantile of y . igure 1: Ratios of the tail index estimates’ RMSEs DGP1: Pareto (k=0.20 and y c = Q y (0 . y c = Q y (0 . DGP3: Burr ρ =-2 (k=0.05 and y c = Q y (0 . ρ =-2 (k=0.10 and y c = Q y (0 . . . DGP5: Burr ρ =-2 (k=0.20 and y c = Q y (0 . ρ =-2 (k=0.20 and y c = Q y (0 . Note: Ratio RM SE ( ˆ α c ) /RM SE ( ˆ α ) and Ratio RM SE ( ˜ α ) /RM SE ( ˆ α ). Ratio 1 com-pares the censored estimator with the best estimator ˆ α , while Ratio 2 compares the estimator thatignores the censoring with the best estimator ˆ α . We report the results using the Pareto and theBurr distributions as DGP. .2 Imputing mean wages To provide further insights on the usefulness of the procedure introduced in this paper weprovide next an analysis of the performance of the different methods for imputing meanwages. In specific, we compare the following three methods:i) the Pareto-imputed mean wage above the top-code y c ,ˆ τ ( y c ) = ˆ α ˆ α − y c (3.3)where ˆ α is the tail index estimate considering an uncensored Pareto distribution asin Section 2 (see e.g. Hill (1975) and Nicolau and Rodrigues (2019) for tail indexestimators) and y c is the top-code threshold;ii) the imputed mean wage based on the approach suggested by Armour et al. (2016)ˆ τ ( y c ) = ˆ α cHill ˆ α cHill − y c (3.4)where ˆ α cHill is a consistent estimate of the tail index parameter computed as in (2.6).Note that τ ( y c ) := E ( y i | y i > y c ) ;iii) the imputed mean wage based on the method introduced in this paperˆ τ ( y c ) = ˆ α ( x i )ˆ α ( x i ) − y c (3.5)where ˆ α ( x i ) = exp (cid:16) x (cid:48) i ˆ θ (cid:17) . Note that τ ( y c ) := E ( y i | x i , y i > y c ) . The main difference between ˆ τ ( y c ) and the other two approaches (ˆ τ ( y c ) and ˆ τ ( y c )) isthat in the former the particular characteristics of the individuals whose wage is above thethreshold are taken into account through x i . Interestingly, the estimator ˆ τ ( y c ) correspondsto the optimal mean square predictor because τ ( y c ) is the conditional expectation of y i given x i and y i > y c . This follows from the well known result E ( y i − E ( y i | x i , y i > y c )) ≤ E ( y i − g ( . )) where g ( . ) is any other predictor of y i given y i > y c . It turns out that τ ( y c ) isoptimal only if y i is mean-independent of x i , in which case both τ ( y c ) and τ ( y c ) coincide.Thus, having established the superiority of τ ( y c ), it remains to be shown how muchimprovement is provided by τ ( y c ) compared to τ ( y c ) and τ ( y c ) when computing the meanwage above y c . The following Monte Carlo study tries to answer this question. Our experi-ment is based on the following steps: 12. Select a sample size from n ∈ { , , , , , } ;2. Simulate y i , i = 1 , , ..., n according to a conditional Pareto distribution P ( α ( x i ))where α ( x i ) = exp (1 + 2 x i ) and x i ∼ U (0 , . For each i = 1 , , ..., n simulate α ( x i )and then the corresponding y i ;3. All observations above quantile 0 . 95, which corresponds to y c , are censored, but theiroriginal values are saved for comparison purposes (these values are used to assess theestimators’ predictive precision). The data used for estimation are { w i , i = 1 , , ..., n } where w i = min ( y i , y c ) , from which we estimate ˆ τ ( y c ) , ˆ τ ( y c ) and ˆ τ ( y c ) . 4. The estimators ˆ τ ( y c ) , ˆ τ ( y c ) and ˆ τ ( y c ) are used to predict the imputed mean valueabove the threshold y c . 5. Steps 2 to 4 are repeated 2000 times and the mean square errors (MSE) of ˆ τ ( y c ) , ˆ τ ( y c ) and ˆ τ ( y c ) are computed.Other combinations of α ( x i ) produce essentially the same results as long as α ( x i ) ≥ α ( x i ) = exp (1 + 2 x i ) (note that E ( α ( x i )) = 3 . . However, α ( x i ) should be set so that α ( x i ) > , otherwise the conditionaland marginal expected values do not exist, and consequently none of the above estimatorswill be well defined. The case 0 < α ( x i ) < ˆ α ( x i ) √ y c ).Figure 2 illustrates our results. The lines represent two MSE ratios, Ratio M SE (ˆ τ ) /M SE (ˆ τ )and Ratio M SE (ˆ τ ) /M SE (ˆ τ ), computed over different sample sizes (n = (250, 500,1000, 2000, 5000, 10000)). 13igure 2: MSE Ratios Computed Over Different Sample Sizes . . . . . 250 500 1000 2000 5000 10000nRatio 1 Ratio 2 As can be observed from Figure 2, the ˆ τ ( y c ) estimator introduced in this paper producesthe best results as all MSE ratios are larger than one. The gains are modest (between 1%and 2%) when the sample size is small (n=250), but they increase steadily as the sample sizeincreases. Another conclusion, is that ˆ τ (Armour et al. (2016)) is better than the (naive) ˆ τ estimator that does not accommodate the censoring. In this section, we use censored publicly available CPS data to evaluate how the right tailindex of the US wage distribution has changed over time and how these changes may differacross the characteristics of individuals, occupations and industries. In specific, we show thatthe new tail index estimator introduced provides very rich and detailed insights about theright tail distribution of wages. We also assess the sensitivity of the adjustment of the topcoded wage to changes over time and across the characteristics of individuals. For the empirical analysis the March CPS files from IPUMS for the period between 1992 and2017 are used. The wage measure is top-coded at $1923 between 1989 and 1997, and at $2884between 1998 and 2017. The sample is restricted to workers between 16 to 64 year-old onfull-time full year basis employed during the CPS sample survey reference week (35+ hoursper week, 40+ weeks per year). Following Autor and Dorn (2013) the real weekly wage dataare weighted by the appropriate CPS weight to provide a measure of the full distribution of14eekly wages paid. , Occupations are defined as job task requirements of the US Department of Labor Dictio-nary of Occupational Titles (DOT, 1977) and Census occupation classifications for routine,abstract and manual task classifications (Autor and Dorn (2013)).In Figure 3 we present the distribution of the weekly wages for 1992, 1997, 1998, 2007,2010 and 2017. From 1992 to 2017 the concentration of wages has become more skewed to theright. Between 1992 and 1997 the mass points around the top-code increased and with therelaxing of the top-code in 1998 real values beyond that top-code are potentially observable.The same phenomenon also occurs in the most recent period. In 2017 the mass point aroundthe current top-code used in the CPS data is much larger. Wages are converted to 2012 values using the GDP personal consumption expenditure deflator. Workers in our sample come from outgoing rotation groups 4 and 8 and according to Unicon: Whenthe Outgoing Rotation files are produced, two rotations are extracted from each of the twelve months andgathered into a single annual file. The weights on the file must be modified by the user before giving reliablecounts. Since the final weight is gathered from 12 months but only 2/8 rotations, the weight on the outgoingfile should be divided by 3 (12/4) before it is applied. The earner weight is gathered from 12 months fromthe 2 rotations. Since those two rotations were originally weighted to give a full sample, the earner weightmust be divided by 12, not 3. . - . . . . - . . . (a) 1992 (b) 1997 . - . . . . - . . . (a) 1998 (b) 2007 . - . . . . - . . . (c) 2010 (d) 2017 The means and proportions of workers according to their characteristics, occupations andindustry, for observations above the 80th percentile, are presented in Table 2. In contrast to1992, in 2017 the population in the right tail is older (41.03 years on average in 1992 and43.83 years in 2017), the percentage of women is larger (25% in 1992 increased to 33% in2017), and is about one year more educated (15.20 years of education in 1992 and 16.06 yearsin 2017). 16able 2: Characteristics of individuals: means and proportions YearMean 1992 2017Age 41.03 43.83Female 0.25 0.33Education 15.20 16.06 Race White 0.90 0.83Black 0.05 0.05Other race 0.05 0.12 Marital Status Married 0.83 0.81Married no spouse 0.01 0.02Separated 0.02 0.01Divorced 0.09 0.09Widowed 0.01 0.01Single 0.14 0.16 Occupations Managers 0.72 0.81Administrative 0.10 0.08Low skill 0.01 0.01Craft 0.04 0.01Operators 0.02 0.01Transportation 0.11 0.08 Industry a Agriculture 0.02 0.03Construction 0.05 0.05Manufacturing 0.22 0.14Transports 0.11 0.08Trade 0.11 0.09Finance 0.09 0.10Repair 0.04 0.10Personal 0.28 0.32Public 0.08 0.09Observations 112,960 64,002Observations above Percentile 80 22,485 12,791 a In Appendix C we provide a detailed description of the industry classification. Notes : This Table reports the means for the variables used in the analysis for both1992 and 2017. These statistics were calculated for the right tail (for observations abovepercentile 80). All variables are reported on a scale between 0 and 1 with the excep-tion of age and education which are reported in years. The occupation dummies using6 aggregate occupation groups are based on the International Standard Classification ofoccupations (ISCO) as used in Autor and Dorn (2013). The category Managers includesmanagement, professional, technical, financial sales and public security occupations. Thecategory Administrative consists of routine non cognitive occupations and includes admin-istrative support and retail sales occupations. The category Low-skill includes low-skillservices, such as cleaning, guard, food, health, janitors, beauty, recreation, working withchildren and other personal low-skill services. The category Craft aggregates precisionproduction and craft occupations. The category Operators refers to machine operators,assemblers and inspectors. Finally, the category Transportation includes transportation,construction, mechanics, mining and agricultural occupations. . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Black Other Race . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Female Occ: Administration . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Single Ind: Finance . . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Ind: Personal Ind: Trade Notes : The graphs presented in this figure represent the proportion of individuals in differentcategories for percentiles 0.05, 0.1, 0.15,..., 0.95, computed for the years of 1992, 1997, 2002, 2007,2012 and 2017. .2 Conditional tail index estimation results Table 3 presents the censored and uncensored tail index regression results for 1992 and 2017.A negative (positive) regression coefficient corresponds to a decrease (increase) in the tailindex (ceteris paribus) and hence a larger (smaller) number of extreme values may resultas a consequence of changes in the specific variable associated to such a coefficient. Beforeanalyzing the partial effects, as described in Section 2.2, the behavior of a tail index estimatecomputed as an average of the conditional tail indexes is examined, i.e.,ˆ α t = 1 n n (cid:88) i =1 ˆ α ( x t,i ) , t = 1992 , ..., α ( x t,i ) = exp (cid:16) x (cid:48) t,i ˆ θ t (cid:17) (we only report results for 1992 and 2017, however the tail indexregression estimation results from 1993 to 2016 can be obtained upon request). The tail indexestimate in (4.1), ˆ α t , provides an estimate of the unconditional tail index after consideringthe characteristics of all individuals in the sample for each year.The results in Table 3 show that in general, the estimates based on the method thatignores censored data underestimate the true effects of the variables. This is especially clearin the estimates for 2017 (e.g. female and finance). It is a consequence of the potentialinconsistency of the uncensored estimates as highlighted in the Monte Carlo section above.However, the direction of the impact of the covariates suggested by the uncensored estimationis consistent with the results obtained from the censored tail index regression.Comparing 1992 and 2017 (censored estimation), we generally observed a decrease in theestimates for 2017 (e.g. Transportation and Craft and Precision). In some cases, althoughin general not statistically significant (see e.g. other races, married no spouse, widowed,Transports and Trade), positive estimates in 1992 become negative in 2017 and vice versa.Considering only the statistically significant covariates it can be observed that Female, Black,Divorced, Single, Low Skill, Craft, Operators, Transportation and Public have a positiveimpact leading to a reduction in the probability of individuals with these characteristics beingin the right tail, whereas Age, Education, Construction, Finance and Repair have a negativeimpact, originating an increase of the probability of individuals with these characteristicsbeing in the right tail. 20able 3: Uncensored and Censored Tail Index Regression Results Uncensored Censored1992 2017 1992 2017Constant 2 . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0 . 066 ) (0 . 076 ) (0 . 075 ) (0 . 121 )Age − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ (0 . 001 ) (0 . 001 ) (0 . 001 ) (0 . 001 )Female 0 . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0 . 014 ) (0 . 013 ) (0 . 016 ) (0 . 022 )Education − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ (0 . 004 ) (0 . 004 ) (0 . 004 ) (0 . 006 ) Race Black 0 . ∗∗∗ . 004 0 . ∗∗∗ . ∗∗∗ (0 . 029 ) (0 . 034 ) (0 . 031 ) (0 . 054 )Other race − . ∗∗∗ . − . 031 0 . . 025 ) (0 . 016 ) (0 . 030 ) (0 . 033 ) Marital Status Married no spouse 0 . − . 036 0 . − . . 063 ) (0 . 046 ) (0 . 070 ) (0 . 091 )Separated 0 . 035 0 . 051 0 . 044 0 . . 042 ) (0 . 059 ) (0 . 047 ) (0 . 096 )Divorced 0 . 018 0 . ∗∗∗ . 025 0 . ∗∗∗ (0 . 019 ) (0 . 022 ) (0 . 022 ) (0 . 036 )Widowed 0 . − . 070 0 . − . . 083 ) (0 . 052 ) (0 . 077 ) (0 . 099 )Single 0 . 009 0 . ∗∗∗ . 015 0 . ∗∗∗ (0 . 018 ) (0 . 018 ) (0 . 021 ) (0 . 029 ) Occupations Administrative 0 . ∗∗∗ . 021 0 . ∗∗∗ . . 020 ) (0 . 022 ) (0 . 023 ) (0 . 047 )Low Skill 0 . 098 0 . 093 0 . ∗ . ∗ (0 . 064 ) (0 . 060 ) (0 . 067 ) (0 . 091 )Craft 0 . ∗∗∗ . 069 0 . ∗∗∗ . ∗ (0 . 033 ) (0 . 056 ) (0 . 035 ) (0 . 086 )Operators 0 . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0 . 050 ) (0 . 062 ) (0 . 053 ) (0 . 090 )Transportation 0 . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0 . 023 ) (0 . 030 ) (0 . 025 ) (0 . 045 ) Industry Construction − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ (0 . 039 ) (0 . 042 ) (0 . 047 ) (0 . 071 )Manufacturing 0 . − . 011 0 . 004 0 . . 028 ) (0 . 029 ) (0 . 031 ) (0 . 052 )Transports 0 . ∗∗∗ − . 016 0 . ∗∗ − . . 021 ) (0 . 027 ) (0 . 024 ) (0 . 040 )Trade 0 . − . ∗∗ . − . . 020 ) (0 . 024 ) (0 . 025 ) (0 . 043 )Finance − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ − . ∗∗∗ (0 . 020 ) (0 . 022 ) (0 . 026 ) (0 . 043 )Repair − . ∗∗∗ − . ∗∗∗ − . ∗∗ − . ∗∗∗ (0 . 028 ) (0 . 022 ) (0 . 033 ) (0 . 041 )Personal 0 . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗ (0 . 016 ) (0 . 018 ) (0 . 019 ) (0 . 033 )Public 0 . ∗∗∗ . ∗∗∗ . ∗∗∗ . ∗∗∗ (0 . 022 ) (0 . 024 ) (0 . 025 ) (0 . 040 ) Note: This table reports the tail index regression results for 1992 and 2017. Thefirst two columns present the uncensored results while the last two columns contain thecensored results. The omitted categories are white, married, working as manager and theagriculture industry. Standard errors in parentheses and *, **, *** indicate significanceat the 10%, 5% and 1% significance levels. . . . . Note : This figure reports the yearly right tail index estimates computed fromboth the uncensored and the censored regressions.Figure 5 plots the tail index estimate computed as suggested in (4.1) for each year from1992 to 2017, based on uncensored and censored tail index regression estimates. As dis-cussed above, the uncensored approach misrepresents the true unconditional tail index, as itignores the top-coded wages, leading to the overestimation of the true values of α t (wherethe bias/inconsistency is due to the fact that the top extreme values are simply not includedin the estimation). On the contrary, the censored estimates take the information of the indi-viduals of the top-coded wages, although the true wages are not known, into account. Figure5 shows that the censored estimates of ˆ α t have declined over the last 20 years. In otherwords, this Figure shows that the probability of observing an extreme value today is highercompared to the 90s or even in the more recent past. This finding supports the idea thatupper-tail inequality has increased since the 90s and has become more pronounced over thelast 20 years. Autor et al. (2008) observed that the 90th percentile wage rose by more than55% relative to the 10th percentile between 1963 and 2005, which represents a significantincrease. However, our approach suggests that the increase in inequality found by Autoret al. (2008) may be a lower bound of the true increase in inequality.Not adequately handling the top-coding may also lead to overestimation of the tail indexin a variety of other applications (this has been recognized in e.g. the analysis of returns toeducation by Hubbard (2011), and differences in gender and race by Burkhauser and Larri-more (2009) who already use approaches to accommodate for the fact that the wage data istop-coded). Our procedure provides additional flexibility by allowing researchers to evaluatewhich determinants impact the probability of being in the right tail of the wage distributionand which determinants do not. Thus, permits a more detailed analysis on how inequality22as spread across industries, occupation, gender and other population characteristics.In what follows, we focus on the (conditional) tail index regression, and especially onthe partial effects computed as discussed in section 2.2. Figure 6 groups the partial effectestimates and rearranges them from the lowest effect to the highest. A negative (positive)regression coefficient translates into a positive (negative) partial effect, which is associatedwith an increase (decrease) in the likelihood of having more extreme values (the right tailbecomes (less) heavier).In the discussion that follows on the partial effects of the covariates, whenever we referto an extreme event, we are referring to observations which are larger than the 96th quantile( u = 0 . - - O cc - T r an s po r t a t i on I nd - P ub li c O cc - O pe r a t o r F e m a l e O cc - C r a ft O cc - Lo w S k ill A ge M S - D i v o r c ed M S - S i ng l e M S - S epa r a t ed I nd - P e r s ona l E du c a t i on I nd - M anu f a c t u r i ng R a c e - O t he r r a c e R a c e - B l a ck O cc - A d m i n i s t r a t i v e I nd - T r an s po r t a t i on I nd - T r ade M S - W i do w ed M S - M a rr i ed no s pou s e I nd - R epa i r I nd - C on s t r u c t i on I nd - F i nan c e Note: For a description of the occupations see note under Table 2. A further important contribution of the approach introduced is its use for the imputationof mean wages above the top-code as described in Section 3.2. Some authors use valuesto adjust the top-coded wage which are time-varying and differ by group. For instance,Macpherson and Hirsch (1995) provide separate Pareto estimates according to gender and byyear from 1973 to 2014 using public CPS-Merged Outgoing Rotation Groups (CPS-MORG).These authors indicate that these values increase over time and are higher for men than forwomen (e.g. for 2014 the adjustment coefficient is 2.06 for men and 1.81 for women). Incontrast, our analysis is based on the March CPS (outgoing rotation groups 4 and 8), andon the weekly wages instead of annual earnings, however, we also find evidence in favour ofchanging adjustment parameters.This renders support to the observation that imputed wages above the top-code, based24n a fixed value may lead to misstatement of results, given that this approach considerswages above the top-code to be independent of time, age, gender, race and other personalcharacteristics; as well as industry and occupation.To impute wages above the top-code, we consider the estimate of E ( y i | y i > y c ; x i ) givenby ˆ τ ( x i , y c ) in (3.5). However, for some individuals in the CPS database, especially thosein the highest wage groups, ˆ α ( x i ) can be in the neighborhood of 1, or even lower than 1,which implies that E ( y i | y i > y c ; x i ) does not exist and, therefore, the estimate ˆ τ ( x i , y c ) isinadequate. For these cases, we use the conditional median of y i given y i > y c and x i , whichis 2 / ˆ α ( x i ) y c . In specific, to accommodate all situations (i.e. low and high values of ˆ α ( x i )) wepropose the following estimator,ˆ τ ( x i , y c ) = (cid:40) / ˆ α ( x i ) y c < ˆ α ( x i ) ≤ c ˆ α ( x i )ˆ α ( x i ) − y c ˆ α ( x i ) > c . (4.2)In the empirical application of this statistic we set c = 1 . 5, since using c = 1 may lead toexplosive estimates as ˆ α ( x i )ˆ α ( x i ) − → ∞ as ˆ α ( x i ) → + . Other values of c in the neighborhoodof c = 1 . α ( x i ) > . 5. Thus, for most cases, theestimate ˆ τ ( x i , y c ) coincides with the second branch of (4.2), which is the ˆ τ ( x i , y c ) estimatorin (3.3). Hence, we use ˆ τ ( x i , y c ) to impute wages above the top-code over all individuals ofthe sample across time (see Figure 7).Figure 7 illustrates the estimates of the imputed wages above the top-code computedfrom the different approaches discussed in (3.3), (3.4) and (4.2). The ˆ τ and ˆ τ estimates aresimilar. This result is expected given that for the overall analysis ˆ τ is based on the valuesof ˆ α t computed as in (4.1), which provides an estimate of the unconditional tail index afterconsidering the characteristics of all individuals in the sample. However, in the case whereestimates for a particular group, occupation or industry are considered, the ˆ τ estimates willcertainly be different from the ˆ τ estimates (see next section).25 Figure 7: Prediction of Topcoded WagesWhen the proportion of wages above the top-code is relatively small (as for example,from 1992 to 2005), the difference between ˆ τ , ˆ τ and ˆ τ is relatively small; however, as morewages are located in the top-coded category (as for example, in the years following 2007),the effect of censored data becomes stronger and the bias (underestimation) produced by theHill estimator more pronounced (ˆ τ ( y c )).Figure 8 illustrates the time varying nature of the factor necessary to compute the imputedmean wages. Recall that to overcome the top-coding bias, in the literature, a constant value isfrequently used to adjust the top-coded wages. For instance, Autor and Dorn (2013) considera sample between 1980-2005; Acemoglu and Autor (2011), between 1973-2009; Autor et al.(2008) from 1963 to 2005; Katz and Murphy (1992) between 1963 and 1987; Autor and Dorn(2013) between 1950 and 2005; Lemieux (2006) from 1973 to 2003; and Beaudry, Green andSand (2013) from 1979 to 2011. Figure 8 shows that using a fixed value may have beenadequate for pre-1992 data, but that the adjustment factor has increased over time reachingan overall value around 1.85 in 2017. 26 . . . . . . . . . Figure 8: Top-coded wage adjustment factors between 1992 and 2017 Figure 9 illustrates the difference of the imputed mean values for individuals in the Finance,Repair, Personal and Public industries, as well as for women and women working in thoseindustries. The purpose of these graphs is to further highlight the importance of allowingfor different scaling factors depending on individuals characteristics and industry, but othergraphs considering other characteristics can be plotted using our approach.The first noticeable result is that the imputed wage of individuals decreases when wecompare the wages for individuals in the Finance, Repair, Personal and Public industries.Finance displays the largest and Public the lowest imputed wages of the four industries. Withthe exception of the Public industry, women’s imputed wages are lower for the other threeindustries and this observation also holds when we condition women’s imputed wages on theindustry they are in. A further interesting result is that the imputed mean wages display anincreasing trend over time in all industries, for women and for women in those industries,which is an indication that the adjustment factors used to compute the imputed wages alsochanges over time. 27igure 9: Imputed mean wages between 1992 and 2017 by industry Ind: Finance Ind: Repair Ind: Personal Ind: Public This is further highlighted in Figure 10, where the graphs show the top-coded wageadjustment factors between 1992 and 2017, for different combinations of women working indifferent industries. Womens top-coded adjustment factor is always smaller than the topcodedadjustment factor that would be applied to males in any industry with the exception of public.This implies that the public industry is the less heavy tailed. On top of that women workingin the public industry earn less in the right tail than women in the right tail working in otherindustries.While women working in personal and repair are not earning much more nor much lessthan in other industries we find that women working in finance would need a much higheradjustment factor. This means that this is the industry in which they have been earningmore and this result is reinforced with a clear positive trend between 1992 and 2017.28igure 10: Adjustment factors between 1992 and 2017 by industry to impute topcoded wages . . . . . . . . . . . . . . . . Ind: Finance Ind: Repair . . . . . . . . . . . . . . . . Ind: Personal Ind: Public This paper provides three important contributions to the literature. The first corresponds tothe introduction of a conditional tail index estimator which explicitly handles the top-codingproblem and an indepth evaluation of its finite sample performance and comparison withcompeting methods. The Monte Carlo simulation exercise shows that the method proposedto estimate the tail index performs well in terms of estimation of the tail index and whenused in the imputation of wages above the top-code when the sample is censored, which isan intrinsic feature of the public-use CPS database.Second, evidence is provided which shows that the factor values used to adjust the top-coded wages have changed over time and across the characteristics of individuals, occupationsand industries and an indication of suitable values is proposed. Interestingly, the empiricalresults show that the upper-tail inequality has increased since the 90s and has become morepronounced over the last 20 years.Third, an in depth empirical analysis of the dynamics of the US wage distribution’s righttail using the public-use CPS database from 1992 to 2017 is provided. The application ofthe procedure to the CPS data reveals that individuals working in industries such as finance,29onstruction and repair are the ones with the more extreme wages. Moreover, it is alsoobserved that the biggest increases in the probability of observing an extreme wage between1992 and 2017 was for individuals that are black, married without a spouse, or widowed.Older and more educated workers continued to have a significant probability of observing anextreme wage but there was no relevant change between 1992 and 2017. Women observed apositive increase between 1992 and 2017, but the impact is still towards an increase of alpha(although smaller in absolute values than in 1992) i.e. a decrease in the probability of anextreme wage. Furthermore, it is also noted that in comparison to an individual working in anon-routine cognitive occupation (managers) all occupations observed a positive contributionto observe an extreme value (although smaller in 2017). However, conclusions are differentacross occupations.Our analysis also showed that women working in finance (public) would need a higher(lower) adjustment factor to impute the top-coded wages. Furthermore, we also observe thatthe adjustment factor used to impute top-coded wages should be adapted over time andacross characteristics of the individuals especially when using censored data. References Acemoglu, D. and Autor, D. (2011). Skills, tasks and technologies: Implications for em-ployment and earnings , volume 4 of Handbook of Labor Economics , chapter 12, pages1043–1171. Elsevier.Armour, P., Burkhauser, R. V., and Larrimore, J. (2016). Using the p areto distribution toimprove estimates of top-coded earnings. Economic Inquiry , 54(2):1263–1273.Autor, D. and Dorn, D. (2013). The growth of low-skill service jobs and the polarization ofthe US labor market. The American Economic Review , 103(5):1553–1597.Autor, D., Katz, L., and Kearney, M. (2006). The polarization of the US labor market. TheAmerican Economic Review , 96(2):189–194.Autor, D., Katz, L., and Kearney, M. (2008). Trends in US wage inequality: Revisioning therevisionists. The Review of Economics and Statistics , 90(2):300–323.Autor, D. H. (2019). Work of the Past, Work of the Future. AEA Papers and Proceedings ,109:1–32.Bernstein, J. and Mishel, L. (1997). Has wage inequality stopped growing? Monthly LaborReview , pages 3–16.Bordley, R., McDonald, J., and Mantrala, A. (1995). Something new, something old: para-metric models for the size distribution of income. Journal of Income Distribution , 6:91–103. 30urkhauser, R. V., Butler, J. S., Feng, S., and Houtenville, A. J. (2004). Long term trendsin earnings inequality: what the CPS can tell us. Economics Letters , 82(2):295–299.Burkhauser, R. V. and Larrimore, J. (2009). Using internal CPS data to reevaluate trendsin labor-earnings gaps. Monthly Labor Review , 132(8):318.Card, D. and DiNardo, J. E. (2002). Skill-biased technological change and rising wage in-equality: Some problems and puzzles. Journal of Labor Economics , 20(4):733–783.Feng, S., Burkhauser, R. V., and Butler, J. S. (2006). Levels and long-term trends in earningsinequality: Overcoming current population survey censoring problems using the GB2distribution. Journal of Business & Economic Statistics , 24(1):57–62.Goos, M. and Manning, A. (2007). Lousy and lovely jobs: The rising polarization of work inBritain. The Review of Economics and Statistics , 89(1):118–133.Goos, M., Manning, A., and Salomons, A. (2014). Explaining job polarization: routine-biasedtechnological change and offshoring. The American Economic Review , 104(8):2509–2526.Hall, P. (1982). On some simple estimates of an exponent of regular variation. Journal ofthe Royal Statistical Association (Series B) , 44:37–42.Harrison, A. (1981). Earnings by size: A tale of two distributions. The Review of EconomicStudies , 48(4):621–631.Hill, B. (1975). A simple general approach to inference about the tail of a distribution. TheAnnals of Statistics , 3:1163–1174.Hubbard, W. (2011). The phantom gender difference in the college wage premium. Journalof Human Resources , 46(3):568–86.Jensen, S. and Shore, S. (2015). Changes in the distribution of earnings volatility. Journalof Human Resources , 3(50):811–836.Kalleberg, A. L. (2011). Good jobs, bad jobs: The rise of polarized and precarious employmentsystems in the U nited S tates, 1970s-2000s . Russell Sage Foundation.Katz, L. and Murphy, K. (1992). Changes in relative wages, 1963 − The Quarterly Journal of Economics , 107(1):35–78.Larrimore, J., Burkhauser, R., Feng, S., and Zayatz, L. (2008). Consistent cell means fortop coded incomes in the public-use march CPS (1976-2007). Journal of Economic andSocial Measurement , 33(2-3):89–128.Lemieux, T. (2006). Increasing residual wage inequality: Composition effects, noisy data, orrising demand for skill? American Economic Review , 96(3):461–498.31evy, F. and Murnane, R. (1992). US earnings levels and earnings inequality: A review ofrecent trends and proposed explanations. Journal of Economic Literature , 30(3):1333–81.Ma, Y., Jiang, Y., and Huang, W. (2019). Tail index varying coefficient model. Communi-cations in Statistics - Theory and Methods , 48(2):235–256.Macpherson, D. A. and Hirsch, B. T. (1995). Wages and gender composition: Why dowomen’s jobs pay less? Journal of Labor Economics , 13(3):426–471.Mishel, L., Bernstein, J., and Shierholz, H. (2013). The State of Working America . Ithaca,NY. Cornell University Press, 12th edition edition.Nicolau, J. and Rodrigues, P. M. M. (2019). A new regression-based tail index estimator. The Review of Economics and Statistics , 101(4):667–680.Parker, S. (1999). The generalized beta as a model for the distribution of earnings. EconomicsLetters , 62:197–200.Piketty, T. and Saez, E. (2003). Income inequality in the United States, 19131998. TheQuarterly Journal of Economics , 118(1):1–41.Wang, H. and Tsai, C.-L. (2009). Tail index regression. Journal of the American StatisticalAssociation , 104(487):1233–1240. 32 ppendix A: Technical Appendix A.1 Proof of Theorem 2.1 For the proof of Theorem 2.1, let us show first that Σ − / y ˙ K c ( θ,y c ) n p −→ 0. Consider thatthe sequence { ( y i , x i ) } is independently distributed. Σ − / y ˙ K ( θ, y c ) = n (cid:88) t =1 (cid:26) I { y ≤ w i We have δ = ¯ F ( y | ∆ x + x ) − ¯ F ( y | x )¯ F ( y | x ) × (cid:16) yy (cid:17) α (∆ x + x ) − (cid:16) yy (cid:17) α ( x ) (cid:16) yy (cid:17) α ( x ) × (cid:34)(cid:18) yy (cid:19) α (∆ x + x ) − α ( x ) − (cid:35) × 100 (A.3)since y = (1 − u ) α ( x ) y thus it follows that, δ = (cid:16) (1 − u ) α (∆ x + x ) α ( x ) − − (cid:17) × . (A.4)Now and given the specification α ( x ) = exp ( φ ( x )) where φ ( x ) is usually of type φ ( x ) = x (cid:48) β it follows that, α (∆ x + x ) (cid:39) α ( x ) + dα ( x ) dx ∆ x = α ( x ) + φ (cid:48) ( x ) exp ( φ ( x )) ∆ x. Therefore α (∆ x + x ) α ( x ) = α ( x ) + φ (cid:48) ( x ) exp ( φ ( x )) ∆ xα ( x ) = 1 + φ (cid:48) ( x ) ∆ x. In conclusion δ = ¯ F ( y | ∆ x + x ) − ¯ F ( y | x )¯ F ( y | x ) × 100 = (cid:16) (1 − u ) φ (cid:48) ( x )∆ x − (cid:17) × . (cid:4) ppendix B: Additional figures on proportion of indi-viduals, according to characteristics, oc-cupation and industry Figure B.1: Marital Status . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Married No Spouse Separated . . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Divorced Widowed . . . . . . . . . Low Skill Craft . . . . . . . . Operators Transports . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Agriculture Construction . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Transport Repair . . . . . . .05 .15 .25 .35 .45 .55 .65 .75 .85 .951992 19972002 20072012 2017 Public ppendix C: Industry classification Table C.1 - Industry classification Agriculture agriculture, forestry, fishing, hunting, mining and utilities; Construction only construction; Manufacturing manufacturing of non-durable and durable goods and wood; Transports transportation, warehousing, utilities electric light; Trade wholesale and retail trade; Finance finance and insurance; Repair business and repair; Personal personal services, entertainment and recreation, professional and re-lated services; Public public administration and armed forces.