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Working Paper 9611
SECTORAL WAGE CONVERGENCE:
A NONPARAMETRIC DISTRIBUTIONAL ANALYSIS
by Mark E. Schweitzer and Max Dupuy
Mark E. Schweitzer is an economist at the Federal
Reserve Bank of Cleveland, and Max Dupuy is a
graduate student at the Woodrow Wilson School
of Public and International Affairs, Princeton University.
For helpful comments and suggestions, the authors
would like to thank seminar participants at the
Humboldt University Conference on Smoothing and
Resampling in Economics and at the Federal Reserve
Banks of Cleveland, Philadelphia, and San Francisco.
Particularly useful suggestions were offered by Eric
Severance-Lossin, J.S. Marron, and Randy Wright.
Correspondence may be sent to Mark E. Schweitzer,
Federal Reserve Bank of Cleveland, Research Department, P.O. Box 6387, Cleveland, Ohio, 44101.
Phone: (216) 579-2014. Fax: (216) 579-3050.
Email: mschweitzer@clev.frb.org.
Working Papers of the Federal Reserve Bank of Cleveland
are preliminary materials circulated to stimulate discussion
and critical comment. The views contained herein are
those of the authors and not necessarily those of the Federal
Reserve Bank of Cleveland or of the Board of Governors
of the Federal Reserve System.
Working Papers are now available electronically through the
Cleveland Feds home page on the World Wide Web:
http://www.clev.frb.org.
December 1996
Abstract
The large shift of U.S. employment from goods producers to service producers has
generated concern over future income distribution, because of perceived large relative
pay differences. This paper applies a nonparametric density overlap statistic to compare
the sectors distribution of full-time, weekly wages at all wage levels. To counter
problematic features of Current Population Survey data--sampling variation at infrequent
wage rates and extensive rounding at common wage rates--we employ nonparametric
density-estimation procedures to isolate the underlying shapes of the densities. The
validity and accuracy of these two approaches when combined is supported by Monte
Carlo simulations. Standard errors and confidence intervals indicate that our results are
statistically significant.
Broad similarity between goods and services wage distributions is found
throughout the period from 1969 to 1993; however, the densities slowly diverge until
1980, after which they tend to converge. By the 1990s, the estimated densities are more
than 95 percent identical. The breadth of this similarity and steady recent convergence
are not easily identified by typical comparison statistics. Furthermore, the wage densities
are most comparable in the central deciles, a finding that disputes the bimodal
characterization of service-sector wages. Two potential explanations for the time pattern
of the overlapping coefficient are considered by forming hypothetical distributions, but
neither of these explanations removes the pattern.
I. Introduction
The dramatic expansion of the share of U.S. workers employed in serviceproducing industries has provoked much controversy.1 Judgments regarding the
desirability of this transformation often imply assumptions about the relative distribution
of wages in the two sectors, and about changes in the nature of the distributions over
time. The shift toward service-producing employment is often credited with changing
certain features of the overall wage distribution -- Bluestone and Harris (1988)stress this
explanation for growing wage inequality. One version of this story contends that the
service-sector wage distribution is somewhat bimodal relative to the goods-producing
distribution.2 Consequently, the growing service sector is blamed for a perceived
replacement of manufacturing and construction jobs at the middle of the overall wage
distribution with low-wage and high-wage service positions.3 Furthermore, differences in
industry wage structures represent an important element in foreign trade explanations of
rising earnings inequality, like Borjas and Ramey (1995).
Academic research on industry relative wage levels has mainly focused on
differentials between narrow industries. While the research in this area (for example,
Krueger and Summers [1987, 1989] and Helwege [1992]) does not necessarily contradict
the results of our analysis because of methodology differences, it reinforced the belief of
1
Barlett and Steele (1992) and Bernstein (1994) are two recent books which warn about wage consequences of the shift away
from goods-producing employment. Newspapers and other popular publications are also a recurring source of similar opinions, for example,
Johnson (New York Times, 1994) and Hoagland (Washington Post, 1993). The 1994 Federal Reserve Bank of Dallas annual report, titled
The Service Sector: Give It Some Respect is fairly representative of the other side of the debate.
2
See Kassab, 1992, p. 4. This view also crops up in newspapers: according to Johnson (New York Times, 1994), As the
Millers [a family supported until recently by manufacturing jobs] gaze into the future...they see an employment landscape shaped like a barbell.
At one end are bankers and lawyers...; at the other end are countermen at fast-food franchises....
3
Barlett and Steele (1992) stress this thesis.
1
large sectoral wage differences. Most analyses of sectoral wage differences focus on
averages (perhaps derived from a regression with controlling variables), which ignores
differences in wage distribution to the extent that they fail to alter the mean difference.4
The statistical tools available to compare two unknown distributions typically rely
either on strong distributional assumptions (for example, equivalence of parameters for a
normal or lognormal distribution), or do not provide estimates of the level of similarity
between nonequivalent distributions (such as the Kolmogorov-Smirnov equality-ofdistributions test). These tests also require exacting confidence levels to reject the
hypothesis that the distributions are distinct when sample sizes reach the thousands of
observations available in the Current Population Survey (CPS). In order to examine the
relative shapes of the sectoral wage distributions, this paper uses a nonparametric
measure of density overlap to examine wage differences between the two sectors over
time. While the statistic has long existed in the literature, it rigorously developed only
with respect to normal distributions and has rarely been applied. We also modify this
statistic in order to identify the locations within the distribution that account for the nonoverlap in each year. The statistical significance of all overlapping statistics in this
analysis is evaluated using bootstrapping techniques.
This statistic is applied both to empirical densities and to smooth densities
estimated using a kernel density estimation procedure. The estimated densities have the
advantage of reflecting the shape of the densities without the large amount of rounding
4
An exception is Lawrence (1984) who compares distributions, in terms of low, middle and high income ranges at two points of
time. This approach would be complicated if extended over the full time period as sectors because of need to choose appropriate cutoffs and
differing business cycles in the sectors. While similar in focus, this analysis is much less general in its coverage and lacks measures of the
statistical significance of the results.
2
evident in the raw data. Rounding lowers the apparent overlap of densities by allowing
economically insignificant variations in pay levels to lead to substantial nonoverlap at
clustered wage levels. Monte Carlo simulations demonstrate two major advantages of
applying smoothing prior to calculating the overlapping coefficient: sharply reduced bias
in the measure and improved small sample estimation. These simulations are based on
controlled samples from commonly used CPS wage data. The combination of these two
techniques enables us to reliably measure broad sectoral wage differences and should be
broadly applicable to other comparisons of wages between groups.
Our results chronicle substantial sectoral wage convergence over the last decade,
and also indicate that overlap has been consistently strongest over the middle quantiles of
the distributions These results reflect secular trends easily distinguished from sampling
variation. We extend these results to show that more narrowly-defined industries and a
hypothetically constant education distribution do not disrupt these trends.
II. The Data
The results in this paper are based on weekly wage data drawn from 25 years of
the March CPS--1970 to 1994. Our weekly wages are constructed from weeks worked
the previous year and total earnings from the previous year, resulting in wage data that
span the period from 1969 to 1993. Annual earnings are corrected for Census Bureau
topcoding procedures that cap reported annual wage and salary earnings at $50,000 to
$199,998, depending on the year.5 While not necessary for most of the analysis in the
5
The topcoding correction assigns all topcoded wage observations the mean of a Pareto distribution truncated at the topcode,
according to the formula reported in Shryock, et al. (1971). The steepness of the distribution prior to the topcode is measured from the 90th
percentile to the topcode.
3
paper, wages are inflated (using the GDP Personal Consumption Expenditures Deflator)
into constant 1993 dollars to allow readers to compare figures across years.
Our sample includes noninstitutional civilian adults who usually worked full time
(at least 35 hours per week) for at least 39 weeks in the previous year. Part-time workers
are not considered, partially because hourly wage data are not available prior to 1985, but
also because we want to consider comparable workers and jobs in each sector. The
differences between full-time and part-time wages, while potentially relevant due to the
higher part-time employment rates in the service sector, reflect a wide variety of factors
(many of them unrelated to employment opportunities) that are not the focus of this
study. The majority of part-time workers choose their hours for noneconomic reasons
(see Dupuy and Schweitzer [1995]). Furthermore, Blank (1990) finds that the lower pay
accorded to part-time positions primarily reflects the workers lower observed and
unobserved skills. We exclude workers listed as reporting less than half of the real 1993
minimum wage to avoid a small number of problematically low wage observations.6
For the sake of comparison with published figures, the difference between sectoral
median weekly wages for our full-time sample are presented in Figure 1. The most
striking feature is the convergence of median wages between 1979 and the early 1990s.
In 1993, the median service job paid $19 per week less than the median goods-producing
job -- down from a 1979 difference of $83. The relatively small differences between
sectors throughout the period are due to focusing on full-time workers.
6
The minimum full-time workweek of 35 hours is used to calculate the weekly earnings implied by this cutoff.
4
However, even for 1993, the wage distributions for the two sectors are statistically
distinguishable from each other. Kolmogorov-Smirnov tests indicate that the null
hypothesis of equal sectoral wage distributions can be rejected with great confidence
(higher than 99.9 percent) for each year in the sample. Furthermore, for both sectors in
each year, Kolmogorov-Smirnov tests reject the hypothesis that wages are distributed
lognormally (again with greater than 99.9 percent confidence).
III. Measuring the Closeness of Distributions
While any number of summary statistics can be used to compare distributions, our
approach focuses on comparisons of probability density functions. The overlapping
coefficient (OVL) compares the frequencies throughout the range of a variable between
two samples. Direct application of the OVL provides an easily interpreted, substantive
measure of the closeness of two samples, drawn from a population of an arbitrary
functional form, when a suitably defined histogram is an adequate representation of the
populations.
The OVL is a straightforward, but seldom used, measure. Bradley (1985) and
Inman and Bradley (1989) promote the use of OVL as an intuitive measure of the
substantive similarity between two probability distributions. Graphically, OVL is the area
where the densities of the two distributions overlap when plotted on the same axes (see
Figure 2). This representation allows a simple hypothesis--that workers in one group are
more likely to earn a particular wage than workers in another--to be expanded across all
possible wage levels.
5
In the discrete case, appropriate for empirical distributions, OVL is formally
defined as
OVL = ∑ min[ f1( X ), f2 ( X )] ,
X
where f 1(X) and f 2(X) are the empirical probability density functions or simply proportions
of the sample. With continuous distributions, OVL is defined analogously with
integration replacing the summation.7 While Inman and Bradleys (1989) development of
OVL focuses on the coefficients estimation and properties assuming normal distributions,
the value of the OVL in this application is due to the fact that OVL is defined without
regard to any distributional assumptions. Furthermore, OVL is invariant to
transformations that are one-for-one and order-preserving (like a price deflator), when
applied to both distributions.
One limitation of OVL was noted by Gastwirth (1975) in the case of income
comparisons: Potentially meaningful changes in income for individuals do not necessarily
alter OVL. In particular, referring again to Figure 2, if one of the observations beyond the
intersection of the densities (v) is given more X (which could be wages), OVL is
unchanged. More generally, for xi the value of X for observation i adding or subtracting D
to is holdings of X such that sign[ f1(xi ) - f2 (xi )] = sign [ f1 (xi + ∆ ) - f2 (xi + ∆ )] leaves
OVL unchanged. While Gastwirth considers this a serious problem for evaluating the
effects of affirmative-action programs on the wages of whites and minorities, in
comparing the wage distributions of industries there is no sense in which it is preferable
for particular workers in one industry to get larger salary increases than in another.
7
Inman and Bradley (1989).
6
On the other hand, we may wish to know what wage ranges cause the distributions
to differ substantially. An example of a hypothesis easily framed in this context is the
following: While wages are quite similar for top earners in both sectors, the service
sector is dominated by good jobs and bad jobs, lacking the midlevel wage opportunities
available in goods production. To address these issues using OVL, we can split OVL
into the overlap associated with a range of wages. Defining qa as the wage rate at the ath
percentile of the full sample (both sectors) and g as a constant percentage, OVL can be
split into quantile ranges:
∑ {min[ f ( X ), f
OVLQ α =
X ∈( qα ,qα + γ ]
1
2
( X )]}
γ
∈[0,1].
For the same reason that OVL is generally unaffected by changes in wages for
specific observations (location doesnt matter), the choice of a does not alter the possible
values that OVLQa may take. In the case where at each wage level between qa and qa+g
the observed frequencies f 1(x) and f 2(x) are always equal, OVLQa equals the sum of the
frequencies of f(x) (the density of the full sample) between qa and qa+g, which by
definition of the percentiles equals g divided by g, or one. The other extreme is defined
by the case where wages in the two sectors are completely disjoint in the range defined by
qa and qa+g; thus the minimum of the two densities is always zero in this range. This
could occur in a variety of ways; for example, when no workers in a sector are paid
wages in the range, or when workers in one sector are paid in even dollar amounts while
the other sector pays in odd dollar amounts.
7
OVL allows intuitive comparisons of the degree of similarity between empirical
distributions across years. OVLQ allows the similarity or dissimilarity to be located
within the distribution of wages.
IV. Nonparametric Density Estimation
In cases where the discrete jumps of frequency (a feature of histograms) are not
an acceptable description of the underlying density, a nonparametric estimate of the
empirical density may be favored. Nonparametric density estimation has been
recommended for exploratory data analysis in the statistics literature because features of
the distribution are often readily visible in the density (Fox [1990] and Révész [1984]).
Nonparametric density estimation can easily be thought of as sophisticated histograms.
The appearance and implicit interpretation of histograms are strongly dependent on the
number of bins. As their binwidth increases (the number of bins is reduced), potentially
interesting details of distribution are lost. However as the binwidth is decreased,
discontinuities due to sampling may arise. Nonparametric density estimation attempts to
strike a balance between these effects when the underlying density is assumed to be
smooth.
In the case of U.S. wage data there are two clear reasons to believe that some
smoothing may be needed: sampling and rounding. The CPS, while an unusually large
survey, is still subject to noticeable sampling errors at the level of detail needed for
empirical density functions. For example, at the fairly common wage of $400 ($10/hour
for 40 hours) only 294 goods-producing workers were surveyed in 1993. Year-to-year
variation in the sample could lead to surprising differences between sectors at a given
8
wage level. If the underlying densities of wages are smooth, then the surrounding wage
rates may yield information that ameliorates this phenomenon.
A very prominent feature of CPS wage data is the high frequency of wage
observations at round numbers. This could be due to recall bias favoring round numbers
on the part of survey respondents or a tendency for employers to round pay to round
numbers. Regardless, the spikes evident in the raw data may not be relevant features for
the purposes of the comparison. For example, a smaller tendency to round in one
industry would alter the measured OVL without implying large or relevant differences in
the underlying wage densities.8
A kernel density estimator smoothes out the discrete jumps in the histogram by
applying a kernel function in place of the frequency of observations at each wage level.
Kernel functions, K(z), are simply probability density functions integrating to one, so a
variety of options exist. Given a selected kernel, the estimated density function is:
)
1 n x − Xi
f K (x) =
∑ K
nh i =1 h
where n is the number of observations in the sample and h is the bandwidth, which
corresponds to half of the range observations assumed relevant for frequency at x. The
choice of a bandwidth can greatly alter the apparent features of the estimated density,
much as the number of bins alters the characteristics of the histogram.
8
Actually, tendencies to round that vary differently over the wage distributions could be equally damaging.
9
A variety of bandwidth selection rules exist in the kernel-density estimation
literature (Jones, Marron, and Sheather, 1994). These rules are typically implementations
of minimizing the Mean Integrated Squared Error,
(
)
MISE(h) = E ∫ f$h - f ,
where f is the actual density estimated by f$h , which is dependent on the bandwidth h.
While this approach has yielded some interesting new bandwidth rules, it does not address
directly the critical need of this analysis--removal of the spikes caused by rounded wage
rates. Further, a single bandwidth is needed for each sector in all years because a given
bandwidth implies a degree of smoothness for the estimated density. OVL estimates can
depend on the degree to which spikes are smoothed, as noted in section II.
In this light, we applied three rules of thumb to provide guidance on what ranges
of bandwidths might be reasonable, but based our final choice on visual inspection. A
critical variable in all bandwidth rules is the number of observations: As observations
rise, the bandwidth goes to zero. Table 1 shows the results of our three rules of thumb for
both sectors in three years: an early year with a small sample with nearly equal sectoral
employment levels (1969); a middle year with a larger sample size, but a smaller goods
sector (1980); and the last year (1993). These rules vary substantially, with Scotts
(1992) oversmoothing rule, designed to be conservative in finding potential modes,
always the largest.
The visually selected bandwidth turns out to be in the middle of the bandwidth
rules of thumb across all of these classes. Specifically, we found that the Gaussian kernel
with a bandwidth of $50 yielded the most complete reduction in rounding without
10
smoothing out local frequency differences in the wage distributions.9 Other bandwidths
were explored with little change in the qualitative results.
Figure 3 shows the remarkable degree to which the CPS data are clustered. The
smooth plot is the Gaussian kernel estimate, which on this scale shows little of the shape
of the kernel (see Figure 7 for a clearer view of this estimate). In this particular case (the
goods sector in 1993), over 77 percent of the weight of the histogram is in spikes above
the smooth density, which represent about 22 percent of the possible wage rates.
Once the densities have been estimated using these techniques, the estimates may
be used to calculate OVL. In this case, OVL is a function of the estimation procedure
and reflects the degree of similarity of the two densities, given underlying densities that
are believed to be smooth. Even without assuming that the population densities are
smooth, the OVL applied to the smooth density indicates the degree of similarity evident
in basic shape of density. This number will typically be higher than the OVL calculated
from the raw sample, due to reduced sampling variation and rounding differences which
can increase the estimated OVL. OVLQ can also be calculated, although the quantile
estimates for the full sample should reflect the same procedure applied to sector
distributions.
V. Diagnostics of the OVL Measures
OVL is a straightforward, visually oriented statistic that we augment with a wellestablished technique for estimating densities; however, the statistical characteristics of
9
Other popular kernels tended to reproduce discrete jumps associated with larger wage clusters at all but the largest bandwidths.
OVL estimates based on these estimated densities would continue to reflect differences in the rates of clustering between the comparison
groups. A similar problem with non-Gaussian kernels was noted by Minotte and Scott (1993) in a similar context.
11
this combined measure as applied to earnings data are not known. We approach this issue
by simulating direct analogues of characteristics of interest using samples based on the
dataset used in this analysis.
Bias of the Overlapping Coefficient
As a statistical measure, OVL is fundamentally biased. This is because any
sampling variation in the two density estimates results in the statistic being strictly less
than one, even when the samples are actually drawn from the same population. Thus,
OVL estimates near 1.0 may indicate that the densities actually are drawn from the same
population. The most obvious solution is to apply an unbiased test like KolmogorovSmirnov, to determine whether the samples are potentially drawn from the same
population. However, this test does not inform us on the closeness.
To address the issue of bias in OVL, we estimate that bias in the context of CPS
earnings data by fabricating samples that are drawn from the same population. Two basic
tests are applied: 1) The actual wage density for one industry is sampled with
replacement to simulate a population with substantial rounding of earnings levels, and 2)
Samples are drawn from a lognormal distribution with the empirical mean and variance of
the wages used in the first test, which eliminates the rounding in the CPS data. These
tests are applied at both large (»25,000 per sector) and small (»10,000-13,000 per sector)
sample sizes. These simulations are repeated a thousand times to estimate the distribution
of bias for each case.
Table 2 presents the results of the simulations for both the OVL as applied to the
empirical density and the estimated OVL along with its quantiles for each scenario. The
12
starkest conclusion of this analysis is the large degree to which OVL as applied to
empirical density (OVL [raw]) is biased away from 1.0. The OVL of the kernel density
estimates (OVL [sm]) is biased much less (1.0 to 1.6 percent on average), but still
noticeably. The simulations underlying Table 2 also indicate that the bias does not vary
substantially relative to its average level in any given sample: For either OVL, the
standard deviation of the bias simulations is always under 0.5 percent. In all cases,
reducing the sample size increases the bias; however, the bias estimates for OVL (sm) are
increased only by about half a percentage point for a sample reduction of approximately
50 percent.
The quantile bias measures indicate that the bias in the estimated density OVL are
concentrated in the tails of the density. These differential biases must be accounted for
when the OVL is broken into OVLQ. These biases blunt one conclusion of our analysis,
but having been recognized, they can be easily accounted for without losing the ability to
address the location of the differences in the densities.
The Role of Sample Size
OVL being calculated at all wage rates implies that reducing even the large CPS
sample can increase the measured overlap. To estimate the role of sample size across a
broad range of samples, simulations on the 1993 data are run for both OVL measures with
sample sizes from 4,907 to 196,270. In the smaller samples, 90 to 10 percent samples
were drawn from both sectors wage distributions, prior to estimating the full set of
overlapping coefficients. A new sample is drawn for each sample size. Larger sample
sizes are created by adding samples drawn with replacement of the size of the original
13
dataset to yield datasets from double to quadruple the size (49,069) of the original 1993
sample. In order to estimate the sampling distributions of the simulations, these
procedures are repeated 100 times.
The results of the sample-size simulations are shown in Figure 4. OVL (sm) is the
mean of the simulations on the OVL of the estimated density, and OVL (raw) is the mean
of the simulations for the empirical density. The dotted lines indicate one-standarddeviation bounds around the simulation means. The key conclusion is that OVL (sm) are
roughly constant at any sample size. On the other hand, OVL applied to the raw data
deteriorates rapidly. A 90 percent reduction in the sample lowers the OVL estimate from
the raw data from almost 0.85 to 0.69, while the OVL of the estimated densities declines
only a third as much, from 0.95 to 0.93. This characteristic is very important, because the
CPS sample size has nearly doubled over the period, and some of the comparisons that
will be made in the extensions section involve even smaller samples. Both statistics are
only slightly affected by expanding their sample size through sampling with replacement.
VI. The Evidence for Convergence since the Early 1980s
The substantial amount of wage variation in any year is evident from the
estimated densities, shown in Figures 5 to 7. Further, while the distributions of earnings
have changed over time, the two sectors earnings distributions have generally been
reasonably similar. The most notable distinction between the wage distributions is the
higher frequencies of goods workers in the range from $700 to $1,100 in 1980. The
sectoral densities are visually more similar in 1969 and 1993 than in 1980. These
14
qualitative dimensions of relative earnings, while potentially derivable in a more
traditional approach, are obvious from the estimated density.
Quantifying these comparisons with OVL allows fine distinctions to be identified
and the statistical reliability of these observations to be tested. As section III showed,
both OVL and OVLQ estimates are bounded by zero and one. The perfect overlap bound
of one is approached in certain ranges of Figure 7, but can only be obtained if the
employment frequencies in the two sectors are identical at every wage rate. Because
both the calculated statistics and the bootstrapped confidence intervals reflect these
bounds (they never equal one), it is useful to keep a level of effective equivalence in
mind. Given estimated distributions that reflect only variation in the location and the
general shape of the distributions, this level should be high: we will use 0.95 (nearly
equivalent) and 0.98 (effectively equivalent). These numbers imply that, for wages in the
relevant range, 100 workers in the more prevalent sector would typically be matched with
at least 95 or 98 workers in the other. It is helpful to keep cutoffs (though not necessarily
ours) in mind, but the actual estimates are, of course, reported.
While the nonparametric density estimates do not alter the basic character of the
wage distributions, they do significantly alter the implied OVL. Figure 8 shows that the
gap between OVL (sm) and OVL (raw) is substantial, sometimes exceeding 0.1. As noted
above, sampling variation and differences in rounding would tend to increase the OVL
measured in raw data. The other factor in the gap between the two measures is the
summarization of wages implied by the smooth density. To counter the potential problem
of variation in smoothness driving our results, we have also varied the parameters which
affect the smoothness and found similar qualitative results. It should be noted that the
15
estimated densities do show notable features after smoothing, and that the estimated
densities are easily rejected as normal or lognormal.10
The upward trend in OVL since around 1980 is visible in either OVL (sm) or OVL
(raw), although the estimated densities show more convergence. That these trends are
statistically significant can easily be verified in the first two columns of Table 3. The
standard errors derived from a thousand repetitions of the bootstrapping algorithm
described in the appendix are reported in the parentheses for each of the statistics. The
standard errors for both of the OVLs of both the empirical and estimated densities are
quite small--generally less than 0.005; thus, the larger changes of both OVLs are typically
statistically significant. Unfortunately, the bootstrapped standard errors cannot be taken
to imply exact hypothesis tests in this case. One bias already discussed and estimated is
the degree to which the OVL estimates differ from 1.0 when the populations are, in fact,
identical. This bias is not picked up in the bootstrap because each bootstrap sample yields
estimates which also have the same problem. The other bias to be concerned with is the
tradeoff between estimator variability and bias in kernel-density estimates. While this
bias is also picked up by all bootstrap samples, the OVL (raw) estimates give us reason to
suspect that this bias is small, because their standard-error estimates should overstate the
ideal smoothed density errors by virtue of being undersmoothed. Given the known bias,
estimated in Table 2, we expect that the confidence intervals reported here are
conservative reflecting the unconstrained side, with no bias adjustment applied to the
mean, and that the standard errors may be somewhat underestimated.
10
While visual features of these estimates appear to violate the parametric densities, we applied both Kolmogorov-Smirnov tests
and a test based on skewness and kurtosis to verify this statement.
16
In the most recent years, OVL (sm) is approaching levels where we could easily
question the importance of the distinction; however, the choice of cutoffs between
substantial and trivial differences depends on personal interpretations. While the
bootstrapped standard errors are useful for characterizing the variability of our estimators,
we apply bootstrapped confidence intervals to test whether these estimates pass our
hypothetical cutoffs.11 The confidence interval approach is favored, because bounded
statistics tend to result in asymmetric estimation errors as the bound is approached. Again
in Table 3, estimated OVLs that exceed, with 90 percent certainty, the 0.95 cutoff are
indicated by one asterisk, 0.98 by two asterisks, and 0.99 by three asterisks. No fulldensity OVLs exceed the cutoffs with this degree of confidence, but they certainly are
getting close. As measured by our bootstrap analysis, the OVL (sm) estimates in 1993
exceed 0.95 with a probability of almost 0.5.
One of the advantages we noted for OVL is that it can be easily split into quantile
components. Table 3 also shows the decile OVLQs for the estimated densities. While
only in recent years has the convergence of wages for the full distributions reached the
nearly identical cutoff, the middle deciles have frequently exceeded this and higher
cutoffs. Even when the wage distributions were most distinct (1980), the sixth and
seventh deciles qualify as at least 95 percent overlapped, with 90 percent confidence.
These decile OLVQ statistics clearly demonstrate that the wage distributions in the goods
and services sectors of the economy have always been closest in the middle ranges,
belying the oft-made criticism that the services provide only high- and low-paid work
11
estimation.
We follow the approach and guidance of Efron and Tibshirani (1993) on applying bootstrap techniques to confidence-interval
17
relative to goods production. The reality is that the frequencies of middle salary deciles in
the two sectors are highly similar in most years.
The growing convergence in wage distribution in the 1980s and 1990s can also be
allocated according to deciles by the same statistics, because the components average to
the overall.12 Comparing 1980 with 1993, virtually every decile is more similar in 1993,
but the largest changes have been in the second through the fourth deciles and in the top
two deciles. These increases put the fourth through eighth quantiles beyond the 95
percent level of comparability. Wage frequencies are substantially different only in the
lowest two deciles, where service-sector jobs continue to be more frequent, and in the
topmost decile.
What wage ranges led to the peak disparity between distributions seen in 1980?
Again, wages were much more similar in the second through the fourth deciles, along with
the top two deciles, in the early 1970s relative to the early 1980s. In the second through
fourth deciles, it is generally service-sector jobs that are more frequent, whereas the
upper deciles have greater frequencies of goods-sector jobs. Thus, the late seventies and
early eighties were a period when the relative frequencies of employment in the two
sectors became more distinct by shifting towards the wages that are viewed as
conventional for each sector. But the surrounding periods show that the more typical
wage patterns in the two sectors might be more equal.
12
The reported statistics do not average exactly, because the discrete approximation implies variability in the realized quantile sizes,
which are adjusted for in the formula.
18
VII. Further Comparisons
The preceding analysis takes an extreme view of wage comparability that runs
counter to regression analysis: Wages reflect a mixture of investments and compensating
differentials that, while not controlled for, are largely offsetting. While this assumption
has allowed the analysis to focus on the full distribution in ways that are not possible in a
regression framework, this technique does not necessitate a complete lack of controls. In
this section, we consider two simple hypotheses that can be analyzed in the same
framework: 1) that the very broad sectors used in the analysis hide the real wage
differences; and 2) that wages are converging because service-sector workers have
pursued more education, which is rising in value.
Narrower Industries
At the limit, it is self-evident that narrower industries should be more distinct:
Wages in transportation equipment (which includes both automobile and airplane
manufacturers) must be and are different from fast food-restaurants. The workers
employed by the industries are clearly different. Nonetheless, comparisons may be made
at the intermediate categories; for example, manufacturing and narrow services.13 This
particular comparison is relevant because much of the sectoral shift has occurred in these
divisions. Manufacturing employment has been shrinking rapidly, while the narrow
services have been among the most rapidly expanding industries.
13
Manufacturing includes both durable and nondurable components. Narrow services includes: Hotels and Other Lodging;
Personal Services; Business Services; Auto Services; Repair Services; Motion Pictures; Amusement and Recreation Services; Health Services;
Legal Services; Educational Services; Social Services; Museums; Membership Organizations; Engineering and Management Services; and
Private Household Employment.
19
Figure 7 shows that these narrower industries have paralleled the development of
the broader sectors.14 After starting at a relatively high overlap (and with more workers in
manufacturing) wages become more dissimilar, until they reach a minimum in 1980. By
the 1990s wages are nearly as similar in these narrower industries as they are in the
broader sectors. The change is all the sharper in the narrow services, because OVL for
the narrower industries started lower in the early years. For the sake of brevity we did
not report the quantile estimates, but they also repeat the patterns seen in the broader
sectors: Wage frequencies have typically been comparable in the middle deciles, and the
convergence has occurred in the surrounding deciles.
Education
Formal (that is, reported) education levels are higher in the service sector and
have been rising. This fact, combined with the widely observed rising returns to
education, suggests another interpretation of the convergence. Rising education levels
have pushed up the wages of service-sector workers as workers have chosen more formal
education in lieu of high-paying jobs in goods production. While the structural details of
this description are not easily described in the framework, a modified shift-share analysis
is possible. We can ask, What might wages look like if the distributions in both sectors
reflected the education levels of an earlier base year?15
Without the regression analysis to summarize education returns, the hypothesis
must be built in by adjusting the observed frequencies to the base year frequencies. A
14
1969 is not shown because substantial changes in industry coding disrupt comparisons to 1970 and later at this level of
disaggregation.
15
The groups are: Less than a high school diploma, high school diploma, some college but no four-year degree, four-year college
degree, and some graduate school. We use these rough categories in order to compare education over the entire sample.
20
simple approach is to modify the population weights already used in the CPS to reflect the
education distribution of the base year:
edfri y
edwgt iy = wgt i *
,
y
edfri
where wgti is the CPS supplement weight assigned to the individual, and the education
frequency terms (edfri) refer to the population frequency of the individuals education
level in the base and current years. This reweighting implies an assumption that lower
education levels for an individual result in pay comparable to that of current workers at
that education level. Unlike a regression shift-share analysis, it does not assume that
returns to education can be summarized by a single figure for each education level.
While the hypothesis is limited by its assumptions, the results should indicate the
direction of these effects. Even though the education shifts are large in wage
distributions, altering the composition of the labor force to reflect lower education levels
in both sectors affects wages in the sectors fairly evenly. Only in the latest years does
any real distinction develop between the previously estimated OVL and the OVL
constrained to early education levels (see Figure 10). This startling result negates what
seemed to be a fairly credible hypothesis.
VIII. Conclusion
Wages in the goods- and service-producing sectors are much more comparable
than the existing policy literature suspects. The broad-based similarity of wage
frequencies in the two sectors has not previously been examined; rather, economists have
focused on statistically significant average differences, typically in a regression setting
21
with a variety of controls. For many policy applications these controls may not be
relevant (for example, in estimates of the increase in the tax base implied by recruiting
firms from a particular sector). Similarly, our results suggest that policies intended to shift
employment back to goods production from services will not meaningfully alter the
overall distribution of earnings. In fact our results indicate that there is less and less room
for income inequality in the economy to increase due to the goods sector employing a
smaller fraction of the workforce.
This paper proposes and applies an alternative approach to comparing a variable
in two sub-populations that focuses on the similarity of the frequencies over the full
distribution. While we clearly want to support an approach that does not focus so heavily
on the central tendencies of variables, as both means and regressions tend to do, this is
not to suggest that regressions have little value in comparing variables like wages in
subpopulations. Regressions allow the simultaneous summarization of varied controls
which can become impractical in our approach. Nonetheless, we strongly recommend the
use of our techniques to clarify the nature of differences or the location of diminished
differences between wages in related sectors. Our approach allows detailed, statistical
comparisons without making distributional assumptions which are not generally supported
by wage data.
22
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23
Inman, Henry F. and Edwin L. Bradley, Jr., The Overlapping Coefficient as a Measure
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and Jonathan Leonard, eds. (New York: Basil Blackwell, 1987).
Krueger, Alan and Lawrence Summers, "Effeciency Wages and the Inter-Industry Wage
Structure," Econometrica, 56 (1988), 259-94.
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Review, (Fall 1994), 3-11.
Minotte, Michael C. and David W. Scott, The Mode Tree: A Tool for Visualization of
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Révész, P., Density Estimation, in P. R. Krishnaiah and P. K. Sen (eds.), Handbook of
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Chapman & Hall, 1986).
24
Technical Appendix: Algorithms
The Overlap Statistic by Quantiles
This algorithm is exact, given a rounding factor and a smoothing algorithm. While
exact, the choice of these components can alter the estimates. Larger bin sizes increase
the measures overlap. Smoothing can reduce the impact of the rounding factor by
limiting the discrete jumps that typically occur with greater regularity with narrow bins.
1. Collect data into bins according to the rounding factor, R.
2. Assure that within the range of wages in the full sample, frequencies exist for
each bin for both sectors, by assigning zeroes where necessary.
3. Smooth frequency distributions for both sectors, if desired.
4. Calculate and identify the quantiles associated with each wage bin, from the
weighted sum of the sectoral densities.
5. Calculate the overlap at each wage rate, then sum by quantile and over the full
distribution, according to equation .
6. Adjust quantile overlaps for size variation in the quantiles.
Bootstrapped Standard Errors and Confidence Intervals
We apply simple bootstrapping wherever standard errors or hypothesis tests are
reported for overlap coefficients. Most estimates are constructed from a thousand
bootstrap replications to allow reasonably exact confidence intervals.
1. Resample, with replacement from the original dataset, a bootstrap sample of
equal size.
2. Calculate the overlap statistics (smoothed or raw) from the beginning. Store
the results.
3. Repeat steps 1 and 2, until the replication dataset reaches the desired size.
4. Calculate the standard errors from the standard deviations of this dataset, and
confidence intervals from the percentiles of this replications dataset.
25
Figure 1: Difference Between Goods- and Service-Producing Median Weekly
Wages
90
80
70
CWW93
60
50
40
30
20
10
0
1969
1972
1975
1978
1981
1984
1987
1990
1993
Year
SOURCE: Authors calculations from Current Population Survey data.
Figure 2: Graphic Representation of Overlapping Coefficient
SOURCE: Authors drawing.
Figure 3: Extreme Rounding Reduced by Kernel Density Estimation
SOURCE: Authors calculations from Current Population Survey data.
Figure 4: The Effect of Sample Size on OVL Measures
1
0.95
0.9
OVL
0.85
OVL (sm)
OVL (raw)
0.8
0.75
0.7
0.65
0.6
10
20
30
40
50
60
70
80
Percent of 1993 Sample
SOURCE: Authors calculations from Current Population Survey data.
90
100
200
300
400
Figure 5: 1969 Estimated Wage Densities
SOURCE: Authors calculations from Current Population Survey data.
Figure 6: 1980 Estimated Wage Densities
SOURCE: Authors calculations from Current Population Survey data.
Figure 7: 1993 Estimated Wage Densities
SOURCE: Authors calculations from Current Population Survey data.
0.90
Estimated Density
Raw Data
0.80
0.85
OVL
0.95
1.00
Figure 8: Overlapping Coefficients
1969 1972 1975 1978 1981 1984 1987 1990 1993
Year
SOURCE: Authors calculations from Current Population Survey data.
Figure 9: OVL for Narrower Industries
OVL (Est. Densities)
0.96
0.94
0.92
Broad Sectors
Narrow Industries
0.9
0.88
0.86
0.84
1969
1972
1975
1978
1981
1984
Year
SOURCE: Authors calculations from Current Population Survey data.
1987
1990
1993
Figure 10: OVL When Workforce Education Composition Is Held Constant
0.96
Smoothed Overlap
0.95
0.94
0.93
0.92
Education Groups
Vary
0.91
Education Groups
Constant
0.90
0.89
0.88
1969 1972 1975 1978 1981 1984 1987 1990 1993
Year
NOTE: Base year is 1972.
SOURCE: Authors calculations from Current Population Survey data.
Table 1: Bandwidth Selection Rules
Goods
Number of
Observations
Silvermans
Härdles Better
Scotts Oversmoothing
1969
1980
Services Goods
Services
13702
42.2
49.7
76
15191
41.1
48.4
75.2
19116
42.7
50.3
72.1
36583
31.5
37.1
51.9
Goods
1993
Services
13484
50
58.9
82.4
35644
37
43.6
61
SOURCE: Authors calculations from Current Population Survey data.
Table 2: Bias Simulation Results
Distributions
Lognormal
1994 Goods Sector
Large Sample Small Sample Large Sample Small Sample
Avg. Observations
per sector
OVL (raw)
OVL (sm)
OVLQ (sm)
10
20
30
40
50
60
70
80
90
100
24915
0.862
0.990
9966
0.788
0.984
25000
0.893
0.987
13484
0.880
0.985
0.990
0.993
0.994
0.994
0.994
0.993
0.993
0.990
0.988
0.973
0.984
0.990
0.991
0.991
0.991
0.989
0.987
0.985
0.981
0.955
0.988
0.992
0.991
0.993
0.992
0.992
0.988
0.982
0.985
0.967
0.985
0.990
0.991
0.991
0.990
0.989
0.988
0.986
0.981
0.961
SOURCE: Authors calculations from Current Population Survey data.
Table 3: Estimated Overlapping Coefficients
YR
69
70
71
72
73
74
75
76
77
78
79
80
Raw
Overlap
0.84189
(0.0048)
0.84833
(0.0041)
0.84983
(0.0042)
0.85061
(0.0043)
0.86147
(0.0044)
0.84496
(0.0043)
0.81478
(0.0045)
0.82309
(0.0045)
0.83285
(0.0043)
0.82796
(0.0045)
0.83214
(0.0042)
0.82422
(0.0042)
Estimated
Overlap
0.92242
(0.0055)
0.93989
(0.0044)
0.93113
(0.0047)
0.92256
(0.0049)
0.92575
(0.0049)
0.92177
(0.0049)
0.91748
(0.0048)
0.91464
(0.0048)
0.91376
(0.0048)
0.90666
(0.0047)
0.89732
(0.0047)
0.89642
(0.0045)
First
Decile
0.83134
(0.0147)
0.76956
(0.0140)
0.75886
(0.0150)
0.77289
(0.0148)
0.80608
(0.0152)
0.78113
(0.0154)
0.74071
(0.0145)
0.76109
(0.0147)
0.76308
(0.0147)
0.77577
(0.0138)
0.77139
(0.0137)
0.74909
(0.0137)
Second
Decile
0.90418
(0.0110)
0.91962
(0.0109)
0.90440
(0.0114)
0.90306
(0.0116)
0.90018
(0.0118)
0.87077
(0.0117)
0.88423
(0.0117)
0.87912
(0.0111)
0.86711
(0.0113)
0.86862
(0.0111)
0.84616
(0.0104)
0.83831
(0.0104)
SOURCE: Authors calculations from Current Population Survey data.
Third
Decile
0.93550
(0.0101)
0.97382 *
(0.0096)
0.96332 *
(0.0100)
0.94133
(0.0104)
0.93624
(0.0108)
0.93602
(0.0108)
0.94822
(0.0107)
0.92665
(0.0102)
0.90888
(0.0107)
0.90674
(0.0102)
0.88190
(0.0098)
0.88134
(0.0092)
Fourth
Fifth
Sixth
Decile
Decile
Decile
0.95562
0.98433 *
0.95910
(0.0093)
(0.0052)
(0.0098)
0.99587 *** 0.99153 ** 0.98371 **
(0.0043)
(0.0055)
(0.0068)
0.99556 ** 0.99790 *** 0.98570 *
(0.0058)
(0.0048)
(0.0059)
0.96244
0.98389 *
0.97005 *
(0.0103)
(0.0083)
(0.0068)
0.94873
0.97471 *
0.97313 *
(0.0108)
(0.0094)
(0.0067)
0.96738 *
0.98482 *
0.97244 *
(0.0102)
(0.0079)
(0.0070)
0.95583
0.97444 *
0.97356 *
(0.0107)
(0.0094)
(0.0066)
0.94440
0.96037
0.98508 **
(0.0105)
(0.0107)
(0.0038)
0.93481
0.98297 *
0.97404 *
(0.0104)
(0.0057)
(0.0069)
0.91944
0.94708
0.98896 **
(0.0103)
(0.0107)
(0.0037)
0.91086
0.94671
0.98604 **
(0.0098)
(0.0107)
(0.0031)
0.91747
0.95746
0.99021 **
(0.0097)
(0.0103)
(0.0040)
Seventh
Decile
0.92184
(0.0101)
0.96230 *
(0.0077)
0.95129
(0.0079)
0.93365
(0.0076)
0.93158
(0.0075)
0.94071
(0.0078)
0.94975
(0.0076)
0.95690
(0.0074)
0.95695
(0.0074)
0.96228 *
(0.0079)
0.95834
(0.0070)
0.96256 *
(0.0068)
Eighth
Decile
0.90620
(0.0120)
0.93118
(0.0087)
0.92387
(0.0089)
0.92092
(0.0086)
0.92503
(0.0090)
0.92710
(0.0088)
0.92807
(0.0085)
0.93329
(0.0083)
0.93395
(0.0083)
0.92738
(0.0078)
0.91912
(0.0081)
0.93490
(0.0074)
Ninth
Decile
0.91575
(0.0137)
0.93353
(0.0105)
0.92132
(0.0105)
0.93839
(0.0102)
0.92857
(0.0105)
0.91085
(0.0101)
0.90569
(0.0098)
0.89577
(0.0100)
0.90168
(0.0096)
0.87332
(0.0094)
0.84852
(0.0092)
0.87368
(0.0093)
Tenth
Decile
0.90991
(0.0151)
0.93648
(0.0119)
0.90904
(0.0126)
0.89887
(0.0132)
0.93241
(0.0120)
0.92545
(0.0115)
0.91315
(0.0111)
0.90168
(0.0113)
0.91260
(0.0109)
0.89539
(0.0108)
0.90410
(0.0109)
0.85778
(0.0097)
Table 3 (continued): Estimated Overlapping Coefficients
YR
81
82
83
84
85
86
87
88
89
90
91
92
93
Raw
Overlap
0.82744
(0.0044)
0.82059
(0.0043)
0.82813
(0.0045)
0.84207
(0.0044)
0.83519
(0.0046)
0.84146
(0.0045)
0.85491
(0.0043)
0.84514
(0.0048)
0.85023
(0.0044)
0.84752
(0.0047)
0.85320
(0.0045)
0.84708
(0.0044)
0.84848
(0.0046)
Estimated
Overlap
0.91072
(0.0048)
0.90795
(0.0048)
0.92179
(0.0047)
0.93812
(0.0045)
0.92795
(0.0047)
0.92657
(0.0048)
0.93622
(0.0050)
0.93603
(0.0048)
0.94056
(0.0047)
0.94935
(0.0049)
0.95108
(0.0050)
0.95525
(0.0046)
0.94949
(0.0050)
First
Decile
0.75720
(0.0146)
0.79182
(0.0149)
0.81349
(0.0156)
0.82018
(0.0155)
0.76212
(0.0150)
0.81874
(0.0156)
0.82613
(0.0162)
0.81095
(0.0166)
0.82125
(0.0164)
0.82713
(0.0164)
0.83366
(0.0170)
0.83426
(0.0175)
0.80731
(0.0176)
Second
Decile
0.86065
(0.0113)
0.85896
(0.0109)
0.88049
(0.0121)
0.92411
(0.0122)
0.86795
(0.0119)
0.87455
(0.0122)
0.90135
(0.0131)
0.90585
(0.0131)
0.89311
(0.0129)
0.90458
(0.0131)
0.90985
(0.0128)
0.91591
(0.0135)
0.92038
(0.0139)
SOURCE: Authors calculations from Current Population Survey data.
Third
Decile
0.91322
(0.0104)
0.89329
(0.0102)
0.89926
(0.0111)
0.94633
(0.0107)
0.93105
(0.0114)
0.90842
(0.0114)
0.92847
(0.0116)
0.93325
(0.0125)
0.94566
(0.0122)
0.95985
(0.0125)
0.96489
(0.0124)
0.97210 *
(0.0122)
0.96772
(0.0124)
Fourth
Decile
0.93798
(0.0103)
0.91778
(0.0104)
0.91747
(0.0110)
0.95219
(0.0108)
0.96323
(0.0113)
0.94615
(0.0122)
0.93723
(0.0119)
0.96935
(0.0122)
0.97787
(0.0112)
0.98594
(0.0094)
0.97598
(0.0117)
0.99570
(0.0068)
0.98009
(0.0124)
Fifth
Decile
0.96208
(0.0110)
0.94246
(0.0112)
0.96332
(0.0106)
0.97129
(0.0114)
0.99197
(0.0042)
0.97463
(0.0117)
0.97747
(0.0102)
*
0.99513
(0.0082)
*
0.99000
(0.0089)
*
0.99356
(0.0041)
*
0.99428
(0.0044)
** 0.99606
(0.0044)
*
0.99249
(0.0068)
Sixth
Decile
0.98904
(0.0035)
0.98712
(0.0045)
0.98481
(0.0054)
*
0.98254
(0.0108)
** 0.98937
(0.0055)
*
0.99657
(0.0055)
*
0.99193
(0.0048)
** 0.99061
(0.0099)
*
0.99716
(0.0049)
** 0.98860
(0.0055)
** 0.98465
(0.0058)
*** 0.99642
(0.0047)
** 0.98510
(0.0054)
Seventh
Decile
*
0.96670
(0.0069)
*
0.96985
(0.0069)
** 0.98384
(0.0065)
*** 0.98924
(0.0042)
** 0.98941
(0.0055)
** 0.99441
(0.0050)
*
0.98516
(0.0062)
** 0.99013
(0.0059)
*
0.98335
(0.0061)
*
0.98633
(0.0060)
*
0.98148
(0.0064)
*
0.98676
(0.0055)
*
0.97186
(0.0065)
*
*
*
**
**
**
*
**
*
*
*
*
*
Eighth
Decile
0.93265
(0.0080)
0.93895
(0.0081)
0.96363
(0.0077)
0.95226
(0.0075)
0.95457
(0.0076)
0.94915
(0.0080)
0.96154
(0.0073)
0.95304
(0.0078)
0.97096
(0.0074)
0.97077
(0.0074)
0.98003
(0.0069)
0.97405
(0.0074)
0.97315
(0.0074)
*
*
*
*
*
*
*
Ninth
Decile
0.89831
(0.0096)
0.88389
(0.0093)
0.90463
(0.0089)
0.90992
(0.0090)
0.91087
(0.0089)
0.90549
(0.0089)
0.92595
(0.0090)
0.92855
(0.0094)
0.92345
(0.0086)
0.94375
(0.0085)
0.95068
(0.0082)
0.95241
(0.0082)
0.95660
(0.0083)
Tenth
Decile
0.88825
(0.0104)
0.89454
(0.0106)
0.90463
(0.0089)
0.93242
(0.0097)
0.91744
(0.0106)
0.89662
(0.0118)
0.92616
(0.0100)
0.88192
(0.0126)
0.90172
(0.0115)
0.93207
(0.0110)
0.93415
(0.0131)
0.92845
(0.0120)
0.93899
(0.0131)
@FedFRASER