Federal government wage indexes

To construct federal wage indexes, we must overcome one major hurdle: records from the OPM data must be categorized into industry (see appendix table A-1) and occupation groups (see appendix table A-2) that are consistent with NCS aggregations used for the ECI.¹ The latter is straightforward because the U.S. Bureau of Labor Statistics (BLS) uses a crosswalk classification system to map OPM occupations into the Standard Occupational Classification (SOC) system. The former, in contrast, is more difficult because the OPM data do not contain industry codes. To address this problem, we use the department and agency information in the OPM data and machine-learning tools to match OPM and Quarterly Census of Employment and Wages (QCEW) establishments.² An algorithm is developed to select a unique North American Industry Classification System (NAICS) code for each agency observed in the OPM data. This final mapping yields a desired agency-to-NAICS crosswalk that we use to calculate Laspeyres, Paasche, and Fisher wage indexes for a variety of aggregations.³

Wage index number formulas

We have many index number formulas to choose from, including the commonly used Laspeyres and Paasche indexes and the less commonly used Dutot or Jevons indexes.⁴ For exploratory purposes and brevity, we focus on the Laspeyres, Paasche, and Fisher indexes.

Given wages and employment for periods 0 (base period) and 1 (comparison period), the Laspeyres and Paasche wage index number formulas use a fixed “basket” of jobs (employment) to compute the ratio of total wage costs for period 1 to total wage costs for period 0. The Laspeyres index uses the fixed basket to be period-0 employment, whereas the Paasche index uses the fixed basket to be period-1 employment. These formulas are given by

and

where  and  are the Laspeyres and Paasche indexes,  is hourly wage,  is the expenditure share, and i is job 1, 2, ..., n. The expenditure share is given by

where  is employment, i and j are jobs , and t is period 0,1.⁵ In theory, employers can be expected to substitute away from more expensive workers. Since the Laspeyres index uses a period-0 fixed-employment basket, the Laspeyres index theoretically overstates wage inflation. Conversely, since the Paasche index uses a period-1 fixed-employment basket, the Paasche index theoretically understates wage inflation.

The Fisher wage index is given by the geometric mean of the Laspeyres and Paasche indexes as

Along with the Törnqvist index, the Fisher index is considered to be “superlative,” with a base and comparison period treated symmetrically to better capture labor substitution effects.⁶

Data

BLS has four quarters of OPM data: first quarter of 2019 and second quarter of 2020, 2021, and 2022. For this analysis, we omit the data from the first quarter of 2019 for two reasons. First, 2019 (first quarter) to 2020 (second quarter) straddled the start of the COVID-19 pandemic, which saw large and uncharacteristic changes in the labor market. Second, 2019 (first quarter) to 2020 (second quarter) was a five-quarter period that included two federal salary increases. The data cover workers employed at the end of each quarter. Note that the data are reported to OPM by human resource offices across the federal government and may be subject to some error. If the federal workforce were incorporated into the ECI, data would need to be collected quarterly from OPM.

OPM data include individual federal employees, annual salary, OPM occupation, full-time or part-time status,⁷ grade, agency, city, and state. BLS’s OPM data include workers on military bases (which we exclude) but not postal service employees.⁸ These data do not include any benefit-cost data (e.g., health insurance, retirement, nonproduction bonuses). All salaries are given as annual full-time salaries, so hourly wages are computed by dividing salary by 2,087.⁹ Missing from OPM data are industry data (NAICS codes), so we use QCEW data and some machine-learning tools to construct an agency-to-NAICS concordance.

Also missing from the OPM data are establishment identifiers. So, we identify them by what we observe: agency, city, and state data, which can be used as imperfect proxies for an establishment. When an agency has just a single establishment within a city, city and state work as a perfect proxy. But if an agency has multiple establishments within a city, city and state are imperfect because multiple establishments are identified as a single establishment.

With an agency-to-NAICS crosswalk and a method for identifying establishments, we then map SOC and NAICS codes into occupation and industry groups (sometimes referred to as pseudo-SOC [PSOC] and pseudo-NAICS [PNAICS]). (See appendix tables A-1 and A-2.) Mean wages and total employment are computed for each basic ECI cell (a grouping by PSOC, PNAICS, and job) or subcell (a grouping by PSOC, PNAICS, subcell category, and job). Summary statistics, including employment counts and percentages of total employment from the OPM, are presented in table 1.

Since this analysis is purely exploratory, we do not attempt to reproduce the method for computing the ECI but instead use its basic conceptual framework for computing wage cost indexes for common index number formula.¹⁰ For the ECI, the unit of observation is a quote (such as an establishment, occupation, work status, or grade). These quotes are aggregated into cells consisting of an ownership sector, industry group (PNAICS), and occupation group (PSOC). Cells can be further divided into subcells that may include full- or part-time status, region, division, union status, and so forth.

NAICS codes

Missing from the OPM data are NAICS codes. We construct an agency-to-NAICS crosswalk using QCEW-reported NAICS codes for federal government establishments. The OPM data have standardized, descriptive text for each department and agency. In the QCEW, the department, agency, and NAICS codes are reported individually by each establishment. These reports are subject to variations in establishment practice and can include spelling errors and varying abbreviations. For these reasons, matching the OPM establishments with QCEW establishments is not straightforward.

To construct an agency-to-NAICS crosswalk, we begin by aggregating individual employee data in the OPM data to agency by location. We then match each OPM agency and location with each QCEW establishment by year or quarter, state, and county. For each of these matches, cosine similarities are then calculated for term frequency–inverse document frequency (or TF–IDF) vectorized department descriptions and agency descriptions. This approach essentially amounts to the construction of a cardinal measure of similarity between two vectors. A number of options exist for constructing these vectors for a given match’s descriptions. We have explored bag-of-words unigrams (an unordered list of the individual words from the descriptions) and character n-grams (a contiguous sequence of n characters from a piece of text). We ultimately chose character n-grams because they account for the issue of spelling errors or variations. A key problem with selecting a vectorization strategy is the lack of an objective standard. That is, in the absence of an objective standard, any choice between vectorization strategies possesses some level of arbitrariness.

For a given vectorizer, we use the mean of the cosine similarities for department and agency, weighting by QCEW-reported mean employment and upweighting and downweighting by the relative deviation between employee counts in the OPM establishment-level data and QCEW-reported mean employment. We assume here that larger establishments are more reliable but may also be “punished” for large differences in the reporting of a variable that should be similar. The QCEW department or agency with the best weighted cosine similarity is chosen as the match.

Finally, since each department or agency should uniquely match a NAICS code, we compare the weighted cosine similarity among all establishments for a department or agency and select the NAICS code for the establishment with the best matching weighted cosine similarity. As constructed, the crosswalk is not without flaws, with a mean agency-size weighted score of 0.76 (standard deviation 0.161) and ranging from nearly the worst (0.002) to the best (1.000). The cumulative distribution of cosine similarity scores, weighted by agency size (see chart 1), shows that the bulk of matches are fairly reliable (>0.8), with very few that are clearly unreliable (<0.4). Moreover, the federal government distribution of PNAICS in the OPM dataset roughly matches that for the QCEW data (see chart 2).

For computing exploratory wage indexes, this imperfect crosswalk is sufficient. But to publish indexes using OPM data will require dedicated analyst labor to create a more accurate crosswalk.

Wage index calculations

To compute wage indexes, we first partition the OPM microdata into establishments (department, agency, and city and state) and jobs (occupation, full- or part-time status, and grade).¹¹ Next, we compute average hourly rates and number of employees for each job within an establishment. The establishment-job data are then matched between the second quarter of 2020 and the second quarter of 2021 and between the second quarter of 2021 and the second quarter of 2022. The resulting matched data are partitioned by cell (PNAICS and PSOC) and period. We then calculate weighted average wages and total employment. Finally, we aggregate these data into wage indexes with the use of the Laspeyres, Paasche, and Fisher formulas. To compute subcell wage indexes, we partition the matched establishment-job data by subcell (PNAICS, PSOC, subcell category). Then, we calculate weighted average wages and total employment and aggregate them into subcell wage indexes. Note that for the published ECI, the base period is fixed and all comparisons are relative to the current base quarter (currently the fourth quarter of 2005). In contrast, for each matched pair of OPM datasets (e.g., the first quarter of 2020 to the second quarter of 2021), the base period is the earlier time (e.g., the first quarter of 2020) so that the time series of indexes for each cell and subcell is what is termed “chained.”

Laspeyres, Paasche, and Fisher wage index calculations are shown in tables 1 through 6 for the basic cell aggregation and for a variety of subcell aggregations. We find that our computed rates of inflation are reasonable. Note that the calculations of the Laspeyres, Paasche, and Fisher wage indexes are quite close and, in some instances, equal up to the fourth decimal. This result is similar to other research results.¹² This present research also showed that the expected pattern in which the Laspeyres index exceeds the Paasche index is frequently reversed.¹³ Finally, a comparison of the federal Laspeyres index with the official ECI is given in table 7. Perhaps unsurprisingly, the exploratory federal ECI is more closely aligned with the state and local ECI.

Conclusion

This analysis demonstrates the practicality of using OPM data to compute a federal government wage component of the ECI. Other elements of the ECI may also be feasible if benefit-cost and hours data can be acquired. Given the magnitude of the U.S. federal workforce, its inclusion would expand NCS coverage as well as filling a void in information about federal workers. Although the annually announced federal pay increase provides some information about federal employment cost growth, it is an imprecise indicator—actual cost growth depends on the flow of employees into and out of federal service and the mix of employee tenures. The calculation of a wage or employment cost index would provide BLS data users useful measures of the growth of federal employment costs.

Further exploration of OPM data for use with the NCS will be enhanced by access to benefit-cost data. Even though acquiring benefit-cost data might be infeasible, we believe that the construction of federal wage indexes would prove a valuable addition to the NCS. The addition of the federal workforce to the NCS will require an analyst-validated NAICS crosswalk, which we view to be an attainable goal considering the findings presented in this article.

Appendix: North American Industry Classification System codes by industry and Standard Occupational Classification codes by occupation