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Comparing federal and private-sector wages without logs.

I. INTRODUCTION

Numerous researchers have concluded that workers in the federal government are more highly compensated, on average, than those in the private sector with similar education, experience, and other characteristics. This study reexamines data from the Current Population Survey (CPS) to estimate differences in hourly wages and uses administrative data on federal pay to more accurately impute high earnings that are top-coded in the CPS. Comparing federal workers with workers in the private sector having similar observable characteristics, I estimate that hourly wages were higher in the federal government than in the private sector among less educated workers and lower among those with more education.

What was the difference in average wages between federal and private-sector workers overall? Because of differences in wage dispersion within the sectors for people with similar characteristics, the answer depends critically on the definition of "average." Pooling results from all education levels, I find that the arithmetic mean of hourly wages was about 2% greater among federal workers than among similar workers in the private sector. Most of the previous literature comparing federal and private-sector wages examined differences between the mean log wages of those two sectors (and I also find larger differences in those geometric means), but this study focuses on differences in the arithmetic means of wages, for both practical and theoretical reasons.

As a practical matter, lawmakers have asked what the implications would be for the federal budget if federal workers were paid the same wages as similar workers in the private sector. With the number of federal hours worked held constant, the answer to that question depends on the difference in arithmetic means between the wages of such workers in the federal and private sectors. Differences in mean log wages do not answer that question.

The basic theoretical model of wage determination derived by Mincer (1974) and others leads to a form for wages (Y) as a function of the worker's years of schooling and work experience (X) such that Y = exp(X[gamma]), which Mincer shows fits the data well. But the empirical literature has almost exclusively focused on estimation of the log-linear regression model ln(Y) = X[beta]+ [member of], even though consistent estimates of [beta] are generally not consistent estimates of [gamma]--as discussed by Blackburn (2007). In particular, differences in wage dispersion between two groups of otherwise similar workers (i.e., heteroscedasticity in the log-linear regression models) cause [beta] to be an inconsistent estimate of [gamma], and I find that wages of federal employees were substantially less dispersed than those of similar workers in the private sector.

I measure differences between employees with similar observable characteristics; I do not attempt to address other potential questions of interest, such as what the wages of federal workers would have been if those particular workers had never been employed by the federal government, or what their wages would be if they were laid off from the federal government and moved to the private sector. Those types of questions involve various issues (such as unobserved abilities, selection into and out of the federal sector, and impacts of job loss) that I did not have sufficient data or a credible identification strategy to address.

The wages that this paper focuses on are only one component of compensation. Analyzing differences in overall compensation would involve quantifying the value of the fringe benefits--such as pensions or other employer contributions to retirement savings--provided in each sector. That issue is addressed for federal employees in Falk (2012) and for employees of state and local governments in Gittleman and Pierce (2011a. 201 lb).

II. BACKGROUND AND CONTEXT

The salaries or wages of most federal employees are set based on their rank in a pay schedule, which consists of a series of pay grades and pay steps within those grades. A worker's initial position in the pay schedule is determined by the requirements of his or her job. Jobs on the General Schedule, which covers about 63% of federal employees, are assigned positions based on guidelines established under the Classification Act of 1949. Although that classification system has been modified occasionally, it does not include mechanisms that respond to changes in how private-sector employers distribute compensation across occupations.

Federal employees' movement through the pay schedule is largely determined by tenure. Over time, workers routinely move to higher levels of pay by advancing through the steps of a particular pay grade and occasionally move to higher grades. Employees who perform at high levels can advance more quickly and employees who perform at low levels can be denied step increases, but most federal employees move to progressively higher steps within a particular grade as they are eligible. Although that system ensures that employees in the same job series with similar tenure receive similar pay, it may reduce the flexibility of managers to reward employees who perform at high levels or limit the salaries of employees who are relatively poor performers.

In most years, the salaries at each position in the pay scale are increased by the same percentage within each locality. Under the Federal Employees Pay Comparability Act of 1990, each year the Federal Salary Council (FSC) recommends a general adjustment and a locality-based adjustment to the salaries of workers on the General Schedule. The general adjustment is intended to account for national trends in the salaries of private-sector, state, and local-government jobs as measured by the Employer's Cost Index. The locality-based adjustment is intended to align the average salary of federal employees with the average of private-sector, state, and local-government employees who perform similar jobs in the same metropolitan area. The Federal Salary Council, a function of the executive branch, submits recommendations for those adjustments to the President who can propose an alternative across-the-board adjustment to the Congress. The President has consistently proposed smaller across-the-board adjustments, and Congress eliminated the general and locality-based adjustments for 2011, 2012, and 2013.

While the FSC has persistently recommended large salary increases based on its finding that federal employees are paid less than their nonfederal counterparts, other researchers have found that wages in the federal sector exceed the wages of similar workers in the private sector. Those studies have used log-linear regression analysis and have reported federal wage premiums of 14%-19%. However, research on other topics has demonstrated that that method can produce inconsistent estimates of the percentage differences in the expected values of wages, whereas quasi-maximum likelihood methods provide consistent estimates.

A. Recent Comparisons of Federal and Private-Sector Wages

Two studies have focused on comparing federal and private-sector wages using recent data from the CPS. Biggs and Richwine (2011) found that federal wages were 14% higher, on average, than wages in the private sector when controlling for education, age, sex, race, occupation, part-time employment, marital status, immigration status, location, and the size of the workers employer. Belman and Heywood (2004) did not control for employer size but adjusted for union status. Using older cross sections, which covered from 1997 to 1999, they found that federal wages were 19% higher. Those studies used censored data on earnings, which can cause bias because the Census Bureau's procedure for imputing replacements for censored values does not account for federal employment.

Using different data and a different approach, the FSC regularly compares the salaries paid for federal jobs that are on the General Schedule with the salaries paid for similar jobs in the private sector to inform the President's recommendation for adjustments to federal pay. The FSC found that the average of federal salaries trailed the average of private-sector salaries by 26% in 2011 (Federal Salary Council 2011). The council does not model wages as a function of workers' education, age, or other attributes measured in the CPS. Instead, it compares salaries for federal and private-sector positions that require similar levels of knowledge and entail similar degrees of complexity. Famulari (2002) found that by matching detailed descriptions of positions, the FSC compared federal workers with private-sector workers who have more experience.

Older research examined details of the differences between federal and private-sector pay that may still be relevant. Borjas (2002) and Katz and Krueger (1991) used the CPS to study intertem-poral trends in the distributions of wages for the federal, state, and local government sectors in the context of the rapidly increasing wage dispersion that occurred in the private sector during the 1980s and 1990s. Borjas found that the dispersion of wages, as measured by the difference in the logarithms of wages between the 90th and 10th percentiles of the distribution, had grown more slowly in the federal sector than in the private sector. Katz and Krueger found that the difference between the wages of college- and high-school-educated workers had grown more slowly in the federal sector than in the private sector when adjusted for potential experience, sex, race, part-time employment, and location. They also found that, after adjusting for those characteristics, the dispersion of wages had remained roughly constant in the federal sector although it was growing in the private sector. Borjas, Katz, and Krueger also found that federal--private wage differentials were larger for women than for men.

B. Approaches to Comparing Wages

In the analyses discussed in the previous subsection (except those by the FSC), researchers regressed the natural logarithm of wages on an additive function of workers' measured attributes to control for differences in those attributes. The same approach has been taken in most wage comparisons, including analysis of compensation for employees of state and local governments (e.g., Bender and Heywood 2012). The coefficients in those log-linearized models were estimated using least squares, and the difference in predicted values of log wages between the federal and private sectors, measured in log points, was interpreted as the percentage difference in wages between sectors--usually by exponentiating the difference in log points, implicitly giving the percentage difference in the geometric means of wages for the sectors.

Those studies did not provide further detail about whether the intent was to measure the percentage difference in the expected value of wages or some other characteristic of the wage distributions. However, an older comparison of federal and private-sector wages by Smith (1977) noted that the difference in log wages yields the percentage difference between the geometric means, which generally does not equal the percentage difference in expected values--that is, in arithmetic means. Moulton (1990) and Gyourko and Tracy (1988) explicitly constructed estimates of the percentage difference in the arithmetic means of wages between the federal and private sectors by accounting for differences in the conditional variances of the federal and private-sector wage distributions. Those estimates were constructed under the assumption that the error term in the log-linearized model was normally distributed and based on data that are now over 20 years old.

More recent studies have evaluated the accuracy of estimating the percentage difference in expected values using log-linearized models. The authors concluded that estimation of such models overstated the difference in the expected values of wages between black and white men (Manning and Mullahy 2001), union and nonunion workers (Blackburn 2007), and nurses and non-nurses (Hirsch and Schumacher 2012), because the conditional variance in wages was lower for black men, union workers, and nurses. (1) Similarly, Silva and Tenreyro (2006) found that heteroscedasticity caused log-linearized models to provide inconsistent estimates in the context of international trade flows.

All of those studies suggested using quasi-maximum likelihood estimators (QMLEs) with the exponential form of the model, which leaves the dependent variable untransformed. The term "quasi" is intended to distinguish the subset of maximum likelihood estimators that provide consistent parameter estimates even if the underlying distribution of the data differs from that assumed in the likelihood function (Gourieroux, Monfort, and Trogan 1984). That robustness can be critical when the distribution of the data is in doubt, but QMLEs can be less precise than estimators based on the true distribution. (2)

C. Choosing between Comparisons of Arithmetic or Geometric Means

To see why estimates of arithmetic and geometric means might differ, consider an illustrative comparison of wages for two groups, each of which contains two workers (see Table 1). When the wage dispersion within a group is small, as in group A--where the two workers have wages 20% above and below the group's arithmetic mean--the arithmetic and geometric means for the group are similar. By contrast, in group B--where the two workers have wages 60% above and below the group's arithmetic mean--the arithmetic mean is substantially larger than the geometric means. Thus, greater dispersion leads to the geometric mean of wages for group B being about 20% lower than that of group A even though the arithmetic means are identical.

TABLE 1 Illustrative Comparison of Arithmetic and Geometric Means
(Dollars per Hour)
            Wage of    Wage of  Arithmetic Mean:  Mean of the Logs:
            Worker 1  Worker 2    (1)+(2)/2 (3)   In(1)+In(2)/2 (4)
              (1)        (2)

Group A      30.0       20.0          25.0              3.2

Group B      40.0       10.0          25.0              3.0

         Geometric Mean: (In(1)+In(2)/2) (5)

Group A                24.5

Group B                20.0


Returning to my practical motivation for focusing on arithmetic means, if the federal government had a set of workers paid like those in group A and changed their pay to resemble that of group B to make them comparable with similar workers in the private sector, there would be no effect on the federal budget--even though the mean log wage had been 0.2 log points higher in group A. Thus, the difference in the mean log wage does not inform policymakers of the budgetary consequences of the change in pay.

More generally, consider the difference in the mean log wage for any two groups (i.e., E [ln ([Y.sub.i.sup.A])]--E [In ([Y.sub.i.sup.B])], where [Y.sub.i.sup.A] and [Y.sub.i.sup.B] are wages in groups A and B, respectively). That difference can be decomposed (using a Taylor series evaluated at the expected values of wages) into the difference in the logs of the arithmetic means of wages [mu] and a remainder that depends on the variance [[mu].sub.2] and higher-order central moments:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If the remainder is zero, then the difference in mean log wages equals the difference in the log of mean wages, and thus taking its exponential yields the percentage difference in arithmetic means. The remainder is equal to zero if the shapes of the wage distributions in the two groups are the same, such that all normalized higher-order central moments take on the same values for the two groups (e.g., [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). However, if the s1 apes of the wage distributions differ, then the difference in the geometric means will generally not equal the percentage difference in the arithmetic means.

Blackburn (2007) emphasized testing for heteroscedasticity in choosing whether to use the log-linearized model. Equation (1) illustrates that differences in higher-order moments can also cause large discrepancy between the differences in geometric means those models measure and the differences in arithmetic means. (3)

III. DATA

In this study, I use the CPS to estimate differences in wages between federal workers and private-sector workers with similar observable characteristics. I analyze federal and private-sector wages using data from the Social and Economic Supplement to the CPS, which is administered each March. The March CPS is a nationally representative survey of the civilian noninstitutionalized population, which is conducted annually by the Census Bureau. Respondents are asked about their earnings, sector of employment, and a variety of other attributes of themselves and their employers.

For this analysis, I pool the 2006-2011 cross sections of the March CPS. Because workers report their earnings over the previous year in that survey, the cross sections cover 2005-2010. The cross sections are combined to increase the size of the sample and to allow a comparison of wages that spans periods of economic growth as well as decline. The number of federal workers in the sample used for the analysis is 8,311 and the number of private-sector workers is 211,504.

A. Composition of the Sample

I compare the wages of federal civilian employees whose compensation is directly funded through Congressional appropriations with the wages of private-sector workers who are not self-employed. I do not analyze the wages earned by members of the armed services or by employees of the Postal Service or the Tennessee Valley Authority (TVA). The Postal Service and the TVA do not receive specific appropriations for compensation of their workers; their operations are primarily funded through revenues from services provided. Also, Postal Service employees fall under a different pay scale than other federal employees, and the pay rates for that scale are determined through collective bargaining. To remove those workers from the analysis, I exclude CPS respondents who reported working in the Postal Service or the electric power industry. I also exclude employees of state and local governments, workers over age 64, and self-employed people. The self-employed were omitted because their earnings not only reflect the payments they earn for their labor but also can include the returns on their investments in capita1. (4)

To improve the accuracy of the analysis, I also exclude part-time and part-year workers and individuals who worked multiple jobs. Wages tend to be measured with more error for people who worked less than 35 hours in a usual week or less than 50 weeks during the previous year, because wages are calculated by dividing earnings by the number of hours worked. Thus, those part-time and part-year workers have smaller denominators, which exacerbate errors in the reporting of their earnings. Those workers accounted for about 7% of the hours worked by federal sector and 17% of hours worked in the private sector. For CPS respondents who worked multiple jobs, the sector of employment is only reported for their longest job, and hours worked are only reported as a total for all jobs.

Some respondents do not report their earnings, sector of employment, or other measured attributes. The Census Bureau imputes values for those characteristics for many of those respondents, but the imputation procedure does not account for the respondent's sector of employment. I exclude workers who did not provide their earnings or sector of employment, because the imputed values for those variables do not provide additional information about the relationship between earnings and sector of employment. (5) However, I include workers who had imputed values only for other measured attributes, because their reports of earning and sector provide additional information about the relationship between those variables.

B. Measuring Wages

I calculate hourly wages by dividing the earnings that workers reported for the previous year by the product of the hours they worked in a usual week and the number of weeks they worked in the previous year. Annual earnings include tips, overtime pay, commissions, and bonuses, as well as salaries and are inflated to 2010 dollars based on the Employment Cost Index for wages and salary in private industry. Roemer (2002) found that the averages of annual earnings in the March CPS were similar to averages calculated from the records of the Social Security Administration. However, Lemieux (2006) argues that for workers who are paid by the hour, wages calculated from reports of annual earnings are less precise measures than the direct reports of pay rates available for the outgoing rotation groups of the CPS. That argument is unlikely to present an issue for this research as only a small portion of federal employees are paid by the hour. Moreover, the March CPS has two advantages over the outgoing rotation groups: it includes data on the size of the firms employing workers, and it provides more information on the wages of high earners.

In order to accurately capture differences in high wages between the federal and private sectors, I adjust the values that the Census Bureau had imputed for the 0.7% of federal workers and the 1.2% of private-sector workers who reported earnings over $200,000. The averages that the Census Bureau provides in place of top-coded earnings do not distinguish between the earnings of federal and private-sector workers. (6) I use administrative data that cover over 90% of federal employees to calculate the averages of earnings for federal workers making more than $200,000. (7) As in the Census Bureau's imputation procedure, those earnings are averaged within groups of employees having the same sex, race/origin, and full-time full-year status for each year. Those averages are used in place of the values provided by the Census Bureau for federal employees and are also used to adjust the averages that the Census Bureau provided for private-sector workers. (8) The average of earnings across those groups was $238,220 for federal workers and $432,553 for workers in the private sector.

C. Measuring Sector of Employment

The workforce tabulations in the national income and product accounts indicate that data from the CPS overstate the percentage of the population that works for the federal government, which could bias a comparison of federal and private-sector wages. In the CPS, federal employees accounted for 2.7% of weeks worked by federal and private-sector employees from 2005 through 2010. (9) But according to the national income and product accounts, federal employees accounted for only 1.8% of weeks worked by federal and private-sector employees during those years. That discrepancy suggests that some private-sector employees, perhaps those who worked for private firms that serviced the federal government, misclassified themselves as federal employees in the CPS. However, the average of the earnings for people who reported being federal employees in the CPS was similar to the average of the earnings recorded for federal employees in the administrative data.

IV. CHARACTERISTICS OF WORKERS

The federal workforce tended to be more concentrated in professional occupations--and thus more educated and older--than the private-sector workforce (see Table 2). About a third of federal employees worked in professional occupations, such as the sciences or engineering, whereas a larger portion of private-sector employees worked in blue-collar occupations or retail sales. Professional occupations often require more formal training or experience than do the occupations more common in the private sector. Partly because of that difference, the average age of federal employees was 4 years higher than that of private-sector employees. The greater concentration of federal workers in professional occupations also meant that they were more likely to have a bachelor's degree: 51% of the federal workforce had at least that much education, versus 31% of the private-sector workforce. Likewise, 21% of federal employees had a master's, professional, or doctoral degree, compared with 9% of private-sector employees.

TABLE 2 Composition of the Federal and Private-Sector Workforces

                                     Federal     Private Sector

Average wage (dollars per hour)         32.3            23.6
Average age (years)                     44.9            40.8
  (percentage of workforce)
Highest educational attainment
  No high school diploma                 1.8             9.7
  High school diploma                   18.3            30.9
  Some college, no degree               18.7            18.3
  Some college, associates degree        9.9            10.4
  Bachelor's degree                     30.5            21.6
  Master's degree                       14.1             6.5
  Professional degree                    2.9             1.4
  Doctorate                              3.7             1.2
Occupation
  Management, business, financial       23.6            17.2
  Professional                          32.6            18.2
  Service                               13.5            12.0
  Sales                                  1.6            11.3
  Administrative/office support         15.3            13.9
  Blue collar                           13.3            27.4
Firm size (# of employees)
  Under 10                               0.2            11.5
  10-99                                  0.3            26.2
  100-499                                0.3            16.1
  500-999                                0.1             6.4
  1,000+                                99.1            39.8
Region
  Northeast                             12.6            18.1
  Midwest                               13.6            22.6
  South                                 37.2            34.9
  Washington D.C. metropolitan area     14.0             2.1
  West                                  22.5            22.4
Other demographics
  Female                                43.0            42.4
  Black                                 17.6            10.1
  Hispanic                               9.1            16.2
  Married                               64.4            60.0
  Immigrant                              9.6            18.1
  Not a citizen                          3.1            10.9
  Not in a metropolitan area            10.8            13.2
Observations                           8,311         211.504


The characteristics of employers, as well as of workers, differed between the federal and private sectors. Most federal employees worked for large agencies; the biggest, the Department of Defense, employed about 800,000 civilian workers. Nearly all federal employees worked for entities that had at least 1,000 workers. In contrast, only about 40% of private-sector employees worked for entities with at least 1,000 employees.

The attributes of the federal workforce were more like those of private-sector workers at large firms than those of workers at small firms, because both large firms and federal agencies required a workforce that was more specialized and educated than small firms did. Many federal employees had expertise in specific roles, as over 95% of them worked in agencies that divided tasks among more than 100 occupations. That degree of specialization was not possible for small employers. In addition, only 27% of workers at small firms had at least a bachelor's degree; whereas the proportion of workers with that level of education was greater at large firms (37%) and in the federal government (51%).

V. COMPARING WAGE DISTRIBUTIONS

Before comparing averages of federal and private-sector wages, I juxtapose the wage distribution in the federal sector to wage distribution for similar workers in the private sector. As discussed in Section II, using the log-linearized model will generally lead to inaccurate estimates of the difference in the arithmetic mean of wages if the shapes of those wage distributions differ. Comparisons of wage distributions can also indicate intersector differences in the structure of pay that would be obscured in an analysis of averages.

I compare the wage distributions of federal workers and of private-sector workers who have similar observable characteristics to federal workers, because I am interested in the variation in wages that cannot be explained by differences in the composition of the workforces. Like Biggs and Richwine (2011), I adjust for workers' education, occupation, age, sex, race, marital status, immigration status, location, year, and employer size. (10) I make those adjustments by reweighting the private-sector sample so that it has the same distribution of observable characteristics as the federal-sector sample, an approach developed by DiNardo et al. (1996). Specifically, the CPS weight for each private-sector employee is multiplied by the odds that a worker with his or her characteristics is in the federal sector. The odds are estimated using separate logit models for five major categories of educational attainment: high school diploma or less, some college, bachelor's degree, master's degree, and professional degree or doctorate.

At most levels of educational attainment. I find that the distribution of wages for federal workers is more compressed than the wage distribution for private-sector workers with similar characteristics (see Figure 1). Specifically, the distances between the 25th and 75th percentiles or the 10th and 90th percentiles were smaller in the federal sector than the private sector among workers with bachelor's, master's, or professional degrees (including doctorates). More generally, federal wages were higher at the majority of points in the wage distributions, but the top earners in the federal sector typically received lower wages than their counterparts in the private sector. In particular, federal wages were about half of private-sector wages at the 10th percentile of the distributions for workers with a professional degree or doctorate; but among workers with no more than a high school diploma, federal wages were higher than private-sector wages at the 10th, 25th, 50th, 75th, and 90th percentiles.

VI. COMPARING ARITHMETIC MEANS OF WAGES

The stark differences between the federal and private-sector wage distributions indicate that following the standard approach of estimating a log-linearized wage model would lead to an inaccurate comparison of the arithmetic mean of wages between the sectors.

A. Method

I use the QMLE for the Poisson distribution to estimate the exponential form of the wage model, Y = exp([[gamma].sub.] + [D.sub.[gamma]1] + [X.sub.[gamma]2]). Recall that QMLEs remain consistent even if the data do not follow the assumed distribution and that wages Y have not undergone the log-transformation in the exponential model. In that model, the percentage difference between federal and private-sector wages varies with the workers' characteristics X. (11) In order to address how the government's compensation costs would differ if the average cost of employing federal workers was the same as that of employing similar workers in the private sector, I compare the average of predicted values for federal workers to the average of predicted values for private-sector workers with the characteristics of federal workers. As in the comparison of wage distributions, the characteristics I include in the model are education, experience, occupation, demographics, year, and employer size.

I limit the number of restrictions imposed on the wage model in order to allow for many potential differences between the federal and private-sector wage structures. The consistency of a QMLE only requires that the expected value of wages is correctly specified as a function of the explanatory variables. In order to avoid imposing errant restrictions on the relationships between mean wages and worker characteristics, I estimate separate wage equations for workers in five broad categories of educational attainment. Thus, the wage differentials attributed to worker characteristics can vary by education. Moreover, each model includes a full interaction between those characteristics and sector of employment--that is, the wage differentials attributed to worker characteristics were allowed to vary between the federal and private sectors.

Standard errors for the point estimates are calculated using balance repeated replication as described by Rao (1994). That approach uses replicate weights provided by the Census Bureau to account for the stratified sampling they employ in conducting the CPS and the dependence in sampling across cross sections.

B. Results

On average, compared with private-sector employers, the federal government paid higher wages for workers with low educational attainment but paid lower wages for workers with high educational attainment (see Table 3). The average wage for federal employees overall was about $32 per hour, about 2% higher than the average wage for private-sector workers with the same characteristics. Among workers with a high school diploma or less education, the average wage was 21% higher for federal employees than for private-sector workers with the same measured attributes. In contrast, among workers whose education culminated in a doctorate or professional degree, the average wage was 23% lower for federal employees than for similar private-sector workers. Between those levels of education, the averages of wages in the two sectors were closer to each other. In particular, the average wage for federal employees with a bachelor's degree was about equal to the average wage for similar private-sector employees.

TABLE 3 Comparing Wages by Level of Educational Attainment

                                Average Wages
                             (dollars per hour)

Educational Attainment    Federal   Private-Sector   Percentage
                            (1)     Projections (2)  Difference
                                                     in Averages
                                                         (3)

High school diploma or     23.5*          19.4*         20.9*
less                       (0.4)          (0.2)         (1.8)

Some college               27.1*          23.6*         15.0*
                           (0.4)          (0.3)         (1.6)

Bachelor's degree          35.3*          34.8*           1.7
                           (0.4)          (0.5)          (14)

Master's degree            41.2*          43.4*         -5.2*
                           (0.7)          (0.8)         (1.9)

Professional/doctorate     48.5*          63.2*        -23.3*
                           (1.1)          (2.2)         (2.6)

All levels of education    32.3*          31.6*          2.3*
                           (0.3)          (0.4)         (1.0)

Educational Attainment   Sample Size
                          (Federal;
                           Private)
                             (4)

High school diploma or      1.618:
less                        87.170

Some college                2,339;
                            60,954

Bachelor's degree           2.503;
                            44.380

Master's degree             1,207:
                            13,565

Professional/doctorate        644;
                             5.435

All levels of education     8.311;
                           211.504

Notes: Wages are per hour and include overtime pay, tips, commissions,
and bonuses. Private-sector projections are quasimaximum likelihood
estimates of the wages earned by private-sector workers who match
federal employees in their education, experience, occupations,
demographics, and employer size. Standard errors are in parentheses,
calculated using balance repeated replication.

*p value < .05.


C. Sensitivity of the Results

Model. When using the untransformed wage model, a variety of estimators yield similar estimates of the difference between federal and private-sector wages. Because QMLE is consistent even if the distribution is misspecified, QMLEs for different distributions should give similar estimates as long as the expected value of wages is correctly specified in terms of worker characteristics. Using QMLE with either Poisson, gamma, or normal distributions all result in intersector wage differentials of about 2% (see Table 4).

TABLE 4 Sensitivity of the Wage Differentials to Model

(Percentage Difference in  Without Controls (1)  With Controls (2)
Average Wages)

QMLE with exponential
conditional mean

Poisson                              36.9*           2.3*
                                     (1.3)          (1.0)

Gamma                                36.9*           2.8*
                                     (1.3)          (1.0)

Normal                               36.9*           2.3*
                                     (1.3)          (1.0)

Linear model of wages

Without reweighting                  36.9*           2.2*
                                     (1.3)          (1.0)

With reweighting                       1.5            1.5
                                     (1.3)          (1.0)

Linear model of log wages            51.9*          12.7*
                                     (1.6)          (1.0)

Notes: In column (2), row 1 is the same estimate as in Table 3. Rows
2 and 3 are analogous to row 1 except that they use the gamma and
normal distributions, respectively, for the QMLE. In row 4, wages are
specified as a linear function of worker characteristics, and the
coefficients are estimated by least squares. Row 5 uses the same
specification and estimator as row 4, but the average wage in the
private sector is calculated by averaging the fitted values for the
privatesector observations using the weight discussed in Section V of
the text. In row 6, the log-transformation is used to linearize an
exponential wage model, which is then estimated by least squares, and
the estimate provided in the table is the exponential of the percentage
difference in the averages of log wages. Standard errors are in
parentheses, calculated using balanced repeated replication.

*p value < .05.


Two other approaches that avoid transforming wages, but do not model wages as an exponential function of worker characteristics, yield estimates of intersector wage differentials that are similar to those of the three QMLE methods. The reweighting approach used in Section V to analyze the distribution of wages also produces an estimate of the average intersector wage differential of roughly 2%, as does a model of untrans-formed wages in which differences between the sectors are controlled for using linear regression. The later result suggests that well-known linear regression techniques can be used to accurately calculate percentage differences in arithmetic means if wages are modeled as a flexible function of education and experience. But note that when a more parsimonious linear specification is used--one that imposes a constant derivate of wages with respect to years of education--the estimates of the intersector wage differentials change significantly. (12)

The more traditional approach of estimating the log-linearized model yields wage differentials that are substantially larger. For perspective, the percentage difference between the average federal wage of about $32 per hour and the average private-sector wage of about $24 per hour is 37%, which matches the estimates from the three QMLEs for the level-exponential model when no controls are included for the measured attributes. By comparison, the estimate from the log-linear approach is 52%, which is the percentage difference in geometric means. Once controls are included for the measured attributes, the log-linear approach gives an intersector wage differential of 13%, whereas the three QMLE methods estimate intersector wage differentials of about 2%.

In addition to estimating an untransformed wage model. I am using administrative data to more accurately impute annual earnings in excess of $200,000. When I instead use the values the Census Bureau imputes for top-coded earnings, a substantial gap persists between the intersec-tor wage differential based on the log-linearized model (13.2%) and the one based on the Poisson QMLE (4.0%). The wage differentials estimated from those data reflect that a smaller fraction of federal workers earn over $200,000 but do not account for the lower average of top-coded earnings in the federal sector. (13) Using administrative data to adjust for that lower average reduces the wage differential estimated by the Poisson QMLE from 4.0% to 2.3% (standard error on the difference is 0.5%).14 Researchers who are concerned about top-coded earnings, but do not have access to supplementary data, can impute earnings for those observations by fitting tails to the distributions, similar to Hirsch and Schumacher (2004).

Wage Measure and Sample. To test the sensitivity of the results to the choice of wage measures, I replicate the comparison of averages using reports of weekly earnings from the outgoing rotation groups (ORGs) of the CPS. I limit the sample to workers who were in the ORG during March so that I can control for firm size and adjust top-coded wages using the administrative data. (15) The variances of wages based on weekly and annual earnings were similar within both the federal and private sectors. Moreover, the federal--private wage differential I estimate for ORG wages is not significantly different from the differential I estimate for wages based on annual earnings (see Table Al, columns (5) and (2)).

The estimates of the wage differentials also appear to be robust to the inclusion of part-time and part-year workers. Their wages are probably gauged more accurately by the ORG than the March supplement, because the ORG measure does not rely on the respondents' recollections of the number of weeks they worked during the past year. Including part-time and part-year workers in the ORG sample only increases the intersector wage differentials by half a percentage point, largely because the wage penalties associated with part-time and part-year work are similar between the private and federal sectors.

Controls. Most comparisons of compensation adjust for worker characteristics, such as education and experience, but the literature has not reached a consensus on which job characteristics should be controlled for. Adjusting for differences in the characteristics of workers' jobs is warranted if they proxy for transferable skills, working conditions, or otherwise reflect the wage the workers could earn in the opposite sector (Hirsch, Wachter, and Gillula 1999). I do not control for union membership in the main analysis, because low rates of unionization among private-sector white-collar employees indicates that many unionized federal employees would struggle to find union jobs in the private sector. The inclusion of an indicator for union membership has little effect on the estimate for the intersector wage differential, largely because the rate of union membership is only about 13 percentage points higher in the federal sector (see Table A1. columns (6) and (5)). A worker's occupation is typically viewed as indicative of transferable skills, but Belman and Heywood (2004) find that controlling for detailed occupational classifications in a regression results in about 10% of workers receiving no weight because they are in jobs that are unique to a particular sector. In the main analysis, I use broad occupational groups, but the results change little when controlling for three-digit occupational classifications as over 95% of workers in my sample have intersector counterparts in the same detailed occupation.

APPENDIX

TABLE A1 Sensitivity of Wage Differentials to Wage Measure,
Sample, and Controls

                          (Percentage Difference in Average Wages)

                      March-Supplement       Outgoing-Rotation-Group
                           Wages                     Wages

                    (1)      (2)      (3)      (4)     (5)     (6)

High school         28.1*    20.9*    18.6*   16.5*   15.2*   14.5*
diploma             (1.9)    (1.8)    (1.9)   (4.0)   (4.0)   (4.0)

Some college        21.2*    15.0*    12.6*   14.0*   13.8*   11.9*
                    (1.9)    (1.6)    (2.3)   (3.8)   (3.8)   (3.9)

Bachelor's           8.0*      1.7      2.6   11.3*   11.6*    9.5*
degree              (2.1)    (1.4)    (2.1)   (4.2)   (4.3)   (4.3)

Master's degree       1.4    -5.2*     -2.9    -2.5    -3.1    -3.2
                    (2.0)    (1.9)    (2.0)   (5.5)   (5.7)   (5.6)

Professional/PhD   -17.8*   -23.3*   -25.2*   -12.8   -15.3   -14.0
                    (2.6)    (2.6)    (2.6)   (9.5)   (9.8)   (9.8)

All levels of        8.7*     2.3*      2.0    6.4*    5.9*     5.0
education           (1.2)    (1.0)    (1.2)   (2.8)   (2.8)   (2.8)

Sample

Full-time             Yes      Yes      Yes      No     Yes     Yes
full-year only

Additional
controls(a)

Firm size              No      Yes      Yes     Yes     Yes     Yes

Detailed               No       No      Yes      No      No      No
occupations

Union membership       No       No       No      No      No     Yes

Sample size(b)    219,815  219.815  211,463  31,683  28,474  28,474

Notes: All columns present quasi-maximum likelihood estimates based on
the Poisson distribution. Column (2) contains the same estimates as in
Table 3. Standard errors are in parentheses, calculated using balanced
repeated replication.

(a) All specifications include measures of educational attainment,
experience, broad occupational classifications, and demo?graphics. In
addition, the specification for column (4) includes an indicator of
part-time, part-year employment.

(b)Excludes workers with occupational classifications that are unique
to their sector within the sample.


The comparisons of average wages are somewhat sensitive to whether adjustments are made for the differences in the size of federal and private-sector employers. I control for the size of the firm employing the worker, because jobs are likely to be more specialized in the federal government and at large private firms than at smaller firms, so larger private-sector employers might value the specialized skills of federal workers. However, the higher wages paid by larger private employers may not entirely reflect pay for skills that are transferable between the federal and private sectors. I suspect that controlling for firm size does not completely account for the girth of federal employers, as Brown and Medoff (1989) find that establishment size is also strongly associated with higher wages even when controlling for firm size, and Belman and Heywood (1990) find that federal employees tend to work at larger establishments. I do not test the sensitivity of the analysis to controlling for establishment size, which is no longer recorded by the CPS. When controls are not included for firm size, the average wage for federal workers was 9% larger than the average wage for private-sector workers with similar measured attributes.

VII. DISCUSSION Policymakers and pundits have justified reductions in federal pay based on the budget deficit and studies that find federal employees tend to receive substantially higher wages than their private-sector counterparts. This study demonstrates that when federal wages are measured in accordance with their budgetary implications, the average of federal wages was only about 2% higher than the average for private-sector workers with similar observed characteristics. The wages available for that comparison covered through 2010, and policymakers eliminated the across-the-board increases for federal salaries in 2011, 2012, and 2013, which I suspect will lead to a slight reduction in federal pay relative to the private sector. (16)

Recent policy initiatives have continued to focus on across-the-board changes in federal pay, whereas federal and private-sector pay differs primarily in its distribution across workers within those sectors. Policymakers have proposed eliminating general increases in federal pay for the next couple years, which would have little effect on the distribution of federal pay across workers of varying levels of skill and performance. This analysis shows that less-educated federal employees tend to earn substantially more than their private-sector counterparts, while their more-educated coworkers earn substantially less than similar private-sector workers. I also find that, among workers with the same education and other observed characteristics, federal wages are less dispersed than private-sector wages, which supports the notion that federal pay systems place more limitations on the ability of managers to distinguish pay based on performance.

This study also makes a methodological contribution in demonstrating that the use of the log transformation can yield misleading results. Blackburn (2007) showed that the common practice of applying the log transformation to models of wage determination leads to inconsistent estimation of the model's parameters in the presence of heteroscedasticity. However, even under heteroscedasticity, the transformed model can yield consistent estimates of the difference in the geometric means of wages. A comparison of geometric averages can be more meaningful than a comparison of arithmetic averages, in contexts where it is appropriate to place less weight on more extreme values. But regardless of an employee's position in the wage distribution, a dollar in wages costs his or her employer the same. Thus, a comparison of arithmetic averages is more relevant if addressing the budgetary consequence of public-sector pay policy or other topics regarding the costs employers incur in compensating their workers. I suggest that researchers use a contextual perspective to determine the appropriateness of applying the log transformation.

I use Current Population Survey data from 2005 through 2010 to compare the wages of federal employees and workers in the private sector who have similar observable characteristics. The distribution of wages differed drastically between the federal and private sectors. In particular, I find that federal employees with no more than a high school diploma earned 21% more, on average, than their private-sector counterparts, whereas those with a professional degree or doctorate earned 23% less. Overall, the average of federal wages was about 2% higher than the average wage of similar private-sector workers. Other researchers have found larger differences because they used log-linearized models, which result in comparisons of geometric means. I show that arithmetic means are more relevant in the context of the relationship between a government's compensation policy and its budget. The discrepancy between differences in arithmetic and geometric means occurs because the wages offederal employees were tnuch less dispersed than those of employees with similar characteristics in the private sector (JEL J31, J38, J45)

ABBREVIATIONS

CPS: Current Population Survey

FSC: Federal Salary Council

ORGs: Outgoing Rotation Groups

QMLEs: Quasi-Maximum Likelihood Estimators

TVA: Tennessee Valley Authority

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(1.) Relatedly, BeImam Heywood, and Voos (2002) found that the average log wage of letter carriers for the postal service substantially exceeded the average across all workers in clerical occupations, but that there was substantial variation in average wages across clerical occupations and that the average for most of those occupations was either greater than or not statistically different from that of letter carriers.

(2.) In the statistical software Stata, the glm command executes quasi-maximum likelihood estimation.

(3.) In general, the weights the differences in moments receive in the remainder are the inverse of their order.

(4.) The exclusion of self-employed workers is common, and the CPS excludes them from the earnings questions for the outgoing rotation group.

(5.) About 16% of federal workers and 21% of private-sector workers were excluded from the sample because they had imputed earnings. Hirsch and Schumacher (2004) demonstrate that using those imputed values causes bias in estimators of the coefficient on explanatory variables that are not incorporated in the imputation procedure.

(6.) For the 2011 March CPS, the Census Bureau changed its procedure for protecting the identity of high earners, but I am able to follow the procedure for distinguishing between the average earnings of federal and private-sector workers making over $200,000 that I use for the older cross sections.

(7.) For a description of those administrative data. called the Central Personnel Data File. see Congressional Budget Office. Characteristics and Pay of Federal Civilian Employees (March 2007), p. 2.

(8.) The average for the top-coded earnings of private-sector workers is calculated as a weighted difference between the average of top-coded earnings for all workers and the average of top-coded earnings for federal workers, with the weights based on the portion of top-coded earnings attributed to federal employees. That adjustment removes federal earnings from the average wages used for private-sector workers but leaves the earnings of workers I exclude from the sample. The data do not enable me to estimate the average for the top-coded earnings of workers who were excluded from the sample. The majority of those workers were excluded because they were self-employed.

(9.) For comparability with the national income and product accounts. I calculate the percentage of hours worked by federal employees from a broader sample than was used for the rest of the analysis. The broader sample only excludes employees of state governments. local governments, government-sponsored enterprises, and the armed forces.

(10.) Specifically, my model includes a fourth-order polynomial in potential experience measured as the worker's age less years of education less six; indicators for detailed levels of educational attainment: 9th grade or less, 10th grade. 11th grade, 12th grade. high school diploma, vocational associate's degree, academic associate's degree, bachelor's degree. master's degree, professional degree, and doctorate; a set of 12 indicators representing all combinations of race/origin (Hispanic. Black, and White), sex (males and female). and marital status (married and single); indicators for being an immigrant; being a noncitizen; living outside a metropolitan area: 5 categories for firm size, by number of workers across all the employer's establishments (1-9. 10-99, 100-499. 500-999, and 1,000+); 24 occupational categories: 6 calendar years; and 5 regions (Northeast, Midwest. South. West, and the Washington D.C. metropolitan area). Including an indicator for the D.C. area largely accounts for differences in the sizes of the metropolitan areas in which federal and private-sector workers live.

(11.) For example. exponentiation of the coefficient [[gamma].sub.1] on the indicator of employment D gives the percentage difference in federal and private-sector wages for workers who have values of zero for all of their characteristics.

(12). I imposed that restriction by replacing the indicators for educational attainment with a single variable that measures years of schooling and by estimating a single wage equation instead of separate equations for five broad categories of educational attainment. That linear model yields an intersector wage differential of 4% and education-specific wage differentials ranging from 10% for workers with a high school diploma to -8% for workers with a professional degree or PhD. In contrast, the Poisson QMLE for the exponential form continues to give education-specific wage differentials ranging from about 20% to -20% and a total differential of 2% when restricted to a single equation with education measured in years of schooling, while achieving about a 5% reduction in the standard errors relative to the flexible form of the linear model. Results for the restricted specifications arc available from the author upon request.

(13.) Note that accounting for the larger percentage of federal workers with earnings over $200,000 is crucial to accurate estimation of the intersector wage differential, particularly when comparing the arithmetic averages. If, instead, the workers with top-coded earnings are excluded from the estimation, the Poisson QMLE yields an estimate of 11% and the log-linearized model gives 16%. The details of those results are available from the author upon request.

(14.) Results based on the Census Bureau's imputations are available from the author upon request.

(15.) The Census Bureau imputes weekly earnings of $2.885 ($150,000 per year divided by 52 weeks) for all workers reporting earnings above that threshold. To more accurately measure the wages of high earners. I assume that those workers' weekly earnings exceeded the top-coding threshold of $2,885 per week by the same percentage that their annual earnings exceeded $150,000.

(16.) Federal employees can still receive increases in pay based on tenure and performance. Meanwhile, growth in private-sector pay, as measured by the Employee Cost Index, has been fairly modest, increasing 1.7% from June 2010 to June 2011 and then by an additional 1.8% through June 2012.

JUSTiN R. FALK*

*The author thanks Greg Acs, Andrew Biggs. McKinley Blackburn. Will Carrington. Molly Dahl, Matt Goldberg, Larry Katz, Joseph Kile. Alan Krueger, Alex Mas, David Moore. Brooks Pierce, Jason Richwine, Stephanie Ruiz. James Sherk. Heidi Shierholz. and two anonymous referees for their comments and suggestions.

Falk: Principal Analyst, Micro Studies Division, Congressional Budget Office. Washington DC 20515. Phone 1-202-226-2966, Fax 1-202-226-0207, E-mail just' n.falk @cbo.gov
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