Part II Chapter 4

What Are We Weighting For?

Weighted and unweighted wage regressions agree here. That agreement is informative, and it is not the end of the story.

DataACS 2024 · OLS vs WLS · replicate SEs

Did you use the weights?

The question arrives in the second paragraph of the referee report, and it sounds like a question about software: did you use the weights? You ran a regression on a public-use file. The person weight is a column you left alone, or one you added because a manual said to, and you now have to decide whether to rerun everything, and if the rerun gives a different number, which one is right, and what to say if it does not. This chapter answers the referee with a regression that is easy to care about. A worker weighing a remote offer, a manager setting pay bands for a hybrid team, and a reporter with a headline to write all want the same number: do people who work from home earn more or less than people who commute?

The 2024 American Community Survey (ACS) can answer a version of that question. Among full-time wage and salary workers aged 25 to 64, people who usually worked from home in the survey's reference week earned 0.345 log points more per hour than commuters, about 41% more, in the survey-weighted data. Nobody should publish that number as the answer. Home workers are far more likely to hold a degree: 20.0% of workers with at least a bachelor's degree work from home, against 4.4% of workers without a high school diploma. Adjust for age, sex, and education and the gap is 0.203 log points, about 23%. Add fixed effects for detailed occupation, industry, and state, and it is 0.078, about 8%: an accountant at home set against an accountant in the office. Three defensible numbers, and the choice of comparison, not the arithmetic, separates them. Every one of them describes who earns what; none says what working from home does to pay.

Now the referee's question. The adjusted coefficient is 0.208 log points without survey weights and 0.203 with them, a difference of 0.005. Two conclusions are tempting. The first is that weighting does not matter for this question. The second is that, because the two estimates agree, either one is the average home–commute wage contrast in the population. The first is close to right for the headline, for reasons Solon, Haider, and Wooldridge (2015) set out, and wrong for one group of workers inside it, as this chapter shows. The second is wrong, and Chapter 5 measures by how much in a finer version of the same model.

What are we weighting for?

The ACS does not sample everyone at the same rate. It samples addresses in small counties more heavily than addresses in large ones, some kinds of household answer less often than others, and the Census Bureau's person weight PERWT is built to undo both, then calibrated so the weighted sample matches population totals by age, sex, race, and Hispanic origin (U.S. Census Bureau 2024). For a share, the weight's job is the whole job. Unweighted, 14.0% of the records in this chapter's sample work from home; weighted, 13.7% (95% CI 13.6%–13.8%) of the population they stand for does. The second number is the one a reporter wants.

For the wage gap the question is harder, because "the gap" can mean three different things.

The population's adjusted contrast. In the population of full-time workers, the coefficient on working from home in the best linear predictor of log wages given age, sex, and education. This is a fixed number about the 2024 population, whether or not the linear equation is a true model of wages.
The sample's adjusted contrast. The same coefficient, defined for the sample as drawn. If the ACS over-represents small states and under-represents young renters, this is the projection in a population with that tilt.
The average of many contrasts. The home–commute gap differs from one age-sex-education cell to another. An average of those cell contrasts, weighted by each cell's share of the population, is a third quantity, and a regression does not report it without further work.

Survey weights move the estimator from the second target to the first. They say nothing about the third: a regression averages heterogeneous contrasts with weights of its own choosing, whether or not the records carry PERWT (Chapter 5). And none of the three is causal. Workers and employers decide who works from home, and jobs that allow it differ from jobs that do not in ways the survey does not record. This chapter's target is the first quantity, stated as an equation below, and its question is how much the weights change the estimate of it and what the answer means.

Predict first

If you run the regression below with PERWT as the weight, which of the three quantities do you get? And if the weighted and unweighted coefficients agree, which of the three have you learned something about?

The target is the population linear projection

y_i = \alpha + \beta D_i + X_i'\gamma + \varepsilon_i ,

where $y_i$ is the log hourly wage of person $i$ , $D_i = 1$ if the person works from home (WFH), and $X_i$ holds age, age squared, a female indicator, and four education levels. The coefficient $\beta$ is a well-defined population quantity whether or not the equation is a true model of wages: it is the coefficient on $D_i$ in the population's best linear predictor of $y_i$ . That answers the prediction. The weighted regression estimates the population projection coefficient, which is not, in general, the population's average home–commute contrast. Agreement between the weighted and unweighted fits says something about the first two quantities and nothing about the third.

The sample is the 984,475 ACS person records of wage and salary workers aged 25 to 64 who usually worked at least 30 hours a week, worked at least 40 weeks in the past 12 months, had positive wage income, and were at work in the survey's reference week; 138,148 of them worked from home. Weighted by PERWT ( $w_i$ in this book's notation), they stand for about 105.4 million people."Full-time" here means 30 or more usual hours, a looser line than the common 35. Hourly wage is annual wage and salary income divided by usual weekly hours times the midpoint of the weeks-worked interval, because the ACS asks about weeks only in bands. Wage income is top-coded by state, and 1.4% of records sit at their state's cap.

Analysis Ledger

Do people who work from home earn more? The adjusted contrast (ACS 2024)

Target population: Wage and salary workers aged 25–64 in the 2024 ACS who usually worked 30+ hours a week for 40+ weeks in the past 12 months, had positive wage income, and were at work in the reference week; about 105 million people.
Estimand: The population linear-projection coefficient on working from home (TRANWORK = 80) in a regression of log hourly wage on WFH, age, age², sex, and four education levels. A descriptive adjusted contrast between WFH workers and commuters, not the causal effect of working from home.
Estimator: WLS with PERWT on 984,475 person records; OLS with equal case weights reported alongside.
Explicit weights: PERWT, the ACS person weight: base weight adjusted for noninterview and calibrated to Census population estimates. Equal case weights in the OLS comparison.
Implicit weights: Within education groups, the coefficient averages WFH–commuter contrasts with weights proportional to population share × p(1 − p), where p is the WFH share; high-overlap groups count most (Chapter 5).
Source of randomness: The ACS sample of addresses and the response process. WFH is not assigned by any design.
Variance estimator: Successive difference replication with 80 replicate weights: Var = (4/80)Σ(β_r − β)², df = 79.
Assumptions: TRANWORK codes the primary means of transportation last week, so hybrid workers who mostly commute count as commuters; hourly wages use weeks-worked interval midpoints and state top-coded wage income as released; no causal interpretation.
Numbers reported: 0.203log points
0.208log points
984,475records
13.7%
0.0023log points

Definition

What 'works from home' means in the ACS

The indicator $D_i$ is TRANWORK = 80, "Worked at home," from the ACS question about how the person usually got to work last week. TRANWORK records the primary means of transportation, so a hybrid worker who spends two days at home and three at the office is coded by the commute. Working from home in this chapter means usually working at home in the reference week, not teleworking at all. The Current Population Survey's telework item, TELWRKPAY, counts anyone who teleworked for pay at any time last week, hybrid workers included. The two surveys measure different things and estimate different contrasts, and no reweighting of either reconciles them: Chapter 3's measurement lesson in a new setting. The CPS item is not in this book's data, so this chapter makes no numerical comparison.

The ladder

Predict first

Before looking at Figure 4.1, write down two guesses: how far the ACS person weights will move the adjusted coefficient, and how far occupation and industry controls will move it.

Figure 4.1 walks the coefficient down a ladder of specifications and estimates each rung twice: by ordinary least squares with equal case weights (OLS), and by weighted least squares with PERWT (WLS). The WLS coefficient estimates the population projection. The OLS coefficient estimates the projection for the sample as drawn.

The WFH wage gap shrinks with each control; weighting barely moves it

Coefficient on working from home, log hourly wage, 2024 ACS full-time wage and salary workers aged 25–64

Unweighted
Weighted

Show data

Group	Estimate for	Estimate	95% CI	n
Raw difference	Unweighted (OLS)	0.349	0.345–0.353	984,475
Raw difference	Weighted (WLS, PERWT)	0.345	0.340–0.351	984,475
+ age, sex, education	Unweighted (OLS)	0.208	0.204–0.212	984,475
+ age, sex, education	Weighted (WLS, PERWT)	0.203	0.199–0.208	984,475
+ state	Unweighted (OLS)	0.200	0.196–0.204	984,475
+ state	Weighted (WLS, PERWT)	0.199	0.194–0.203	984,475
+ occupation and industry	Unweighted (OLS)	0.076	0.073–0.080	984,475
+ occupation and industry	Weighted (WLS, PERWT)	0.078	0.073–0.082	984,475

Figure 4.1.Controls shrink the work-from-home coefficient by more than half; weighting barely moves it at any rung.Note: Weighted intervals: 80-replicate SDR, t with 79 df. OLS intervals use household-clustered (CR1) standard errors and the normal critical value. '+ occupation and industry' adds fixed effects for detailed occupation, industry, and state.Source: IPUMS USA, ACS 2024 1-year (SAMPLE 202401), analysis/usa/acs/part_2020_2024.parquet; replicate weights REPWTP1–80 from analysis/usa/acs_repwt/part_2020_2024.parquet

The raw difference is 0.349 unweighted and 0.345 weighted. Adding age, sex, and education cuts it to 0.208 and 0.203. State fixed effects change almost nothing (0.200 and 0.199). Adding fixed effects for 529 detailed occupations and 264 industries, with the state effects kept, removes 62% of the adjusted coefficient, leaving 0.076 and 0.078.

A log-point gap converts to a proportional one as $e^{\beta} - 1$ : about 41% for the raw weighted gap and 23% for the adjusted one. For the raw gap that is the ratio of the two groups' geometric mean hourly wages, which weights low and high earners differently from a ratio of average wages; the levels model below speaks to dollars.

Occupation and industry are not innocent controls. They describe the job, and the job largely settles whether working from home is possible at all. The occupation-adjusted coefficient answers a narrower question, not a truer one, and it conditions on choices made together with the choice to work from home.

Analysis Ledger

The same contrast within occupation, industry, and state

Target population: Wage and salary workers aged 25–64 in the 2024 ACS who usually worked 30+ hours a week for 40+ weeks in the past 12 months, had positive wage income, and were at work in the reference week.
Estimand: The population linear-projection coefficient on WFH after adding fixed effects for detailed occupation, industry, and state: a WFH–commuter contrast among workers with the same recorded job and location, still descriptive.
Estimator: WLS with PERWT and 841 fixed-effect dummies, solved as sparse normal equations.
Explicit weights: PERWT (ACS person weight). Equal case weights for OLS.
Implicit weights: A linear probability model for WFH with additive fixed effects; about 9% of weighted commuters get negative implied weight because their predicted WFH probability is below zero (Chapter 5).
Source of randomness: The ACS sample and response process.
Variance estimator: SDR, 80 replicates, each a full re-solve of the fixed-effect normal equations; df = 79.
Assumptions: Occupation and industry are measured for the main job; controlling for them conditions on outcomes of the same choices that shape WFH, so the contrast is not a structural wage premium.
Numbers reported: 0.078log points
0.076log points
62%

Logs or levels

In the lead example of Solon, Haider, and Wooldridge, the subject of the provenance box below, weighted and unweighted estimates diverged in levels, so the adjusted rung is re-estimated with hourly wages in dollars (Figure 4.2). The gap is $8.62 an hour weighted and $9.04 unweighted, about 24% of commuters' mean hourly wage of $36.09. The weighted–unweighted difference is larger in levels (4.6% of the OLS coefficient) than in logs (2.2%), since levels give the highest earners more leverage, but in the whole sample it stays small.

Logs and levels tell the same story here

Table comparing the work-from-home coefficient in logs and in dollars per hour, unweighted and weighted, raw and adjusted; the implied percentage differences agree within a few points.
Specification	Estimator	Log points	exp(b) − 1	Dollars per hour	Dollars ÷ commuter mean
Raw difference	Unweighted (OLS)	0.349	41.8%	$14.92	40.0%
Raw difference	Weighted (WLS)	0.345	41.3%	$14.38	39.9%
+ age, sex, education	Unweighted (OLS)	0.208	23.1%	$9.04	24.2%
+ age, sex, education	Weighted (WLS)	0.203	22.5%	$8.62	23.9%

Figure 4.2.Logs and levels imply similar proportional gaps, and weighting moves neither by much.Note: Commuter means are weighted for WLS rows and unweighted for OLS rows. Levels use hourly wage in dollars.Source: IPUMS USA, ACS 2024 1-year (SAMPLE 202401), analysis/usa/acs/part_2020_2024.parquet; replicate weights REPWTP1–80 from analysis/usa/acs_repwt/part_2020_2024.parquet

Why the weights barely matter here

Solon, Haider, and Wooldridge give three reasons to weight a regression aimed at a causal or structural parameter. The first is precision. If the error variance differs across observations in a known way, weighting each observation by the inverse of its variance, a precision weight $a_i$ , gives a more efficient estimator; the textbook case is a regression on group averages, weighted by group size. The second is consistency. If the chance of entering the sample depends on the outcome even after conditioning on the regressors, sampling is endogenous, unweighted estimates of the population relationship are inconsistent, and weighting by inverse selection probabilities repairs them (Wooldridge 1999). If sampling depends only on the regressors, it is exogenous, and both estimators are consistent for a correctly specified model, with the unweighted one usually more precise. The third is recovering a population average effect when effects are heterogeneous; they show that weighting alone does not deliver one.

The precision motive has a sharp limit, identified by Dickens (1990). When group averages share a group-level error component, such as a state-wide shock, the variance of a group mean stops shrinking with group size, and weighting by size can make estimates less precise. Neither the motive nor the critique concerns PERWT, which is not a precision weight but a survey weight $w_i$ . Used as a regression weight where it is not needed for consistency, it costs precision. The coefficient of variation of PERWT in this sample is 0.89, so Kish's effective sample for a simple mean, $\nKish/n$ , is 56% of the records (Kish 1965), and the heteroskedasticity-robust variance of the WLS coefficient is 1.65 times that of OLS.

That leaves consistency. If, given working from home, age, sex, and education, the chance of being sampled and responding is unrelated to wages, sampling is exogenous; with a correctly specified model the sample projection and the population projection coincide, and OLS and WLS agree. The ladder says they nearly do. Agreement has a second explanation: the contrast varies across groups, so the model is misspecified, but the groups the weights push up or down have similar contrasts. Either way, the choice of weights is not what moves this coefficient, and the check costs one extra line of code. What agreement does not show is that either coefficient is the third quantity above, the population's average contrast. A regression leans on groups where both home workers and commuters are plentiful, whatever their share of the population. PERWT fixes whom the sample represents; it does not fix how the regression weighs one group's contrast against another's.

When weighting would matter more

The agreement is a finding about this coefficient in this sample, not a rule. Three conditions make weighting matter more.

Inclusion related to the outcome. If the chance of being sampled or of responding depends on wages even given the regressors, the unweighted fit is inconsistent for the population projection (Wooldridge 1999).
Composition that interacts with heterogeneous contrasts. If the contrast differs across groups and the sample's mix of groups differs from the population's, the two estimators average the same contrasts with different weights.
Influential records. A few records with very large weights or high leverage can move a weighted coefficient that the unweighted fit barely notices.

The influence check below finds no dominant group here, and the second condition is the one that surfaces once the sample is split. Whether inclusion depends on wages cannot be checked from the sample alone.

Testing the difference: DuMouchel and Duncan

Predict first

The weighted and unweighted coefficients differ by 0.005 log points. With 984,475 records, will a formal test say that they differ?

DuMouchel and Duncan (1983) turned the comparison into a test. Regress the outcome by OLS on the covariates, the weight, and the weight interacted with each covariate, and test whether the weight terms are jointly zero:

y_i = X_i'b + (\tilde w_i X_i)'c + u_i, \qquad H_0 : c = 0 ,

where $\tilde w_i = w_i/\bar w$ is PERWT divided by its mean and $X_i$ includes the intercept and $D_i$ . The null is the restriction under which the weighted and unweighted estimators share a probability limit. Here the test rejects: F = 75.6 on 8 restrictions with household-clustered standard errors, p < 0.001. The joint test covers every weight term, including those on age, sex, and education, whose coefficients this chapter does not interpret. The term that bears on the work-from-home coefficient is the weight's interaction with $D_i$ , with t = −3.7: workers who carry larger weights show smaller home–commute contrasts. The levels model rejects as well (F = 93.0).

A rejection is a diagnostic, not a verdict (Bollen et al. 2016). With close to a million records the test has the power to detect departures far too small to matter: the two work-from-home coefficients differ by 0.005 log points, 2.2% of the unweighted coefficient and about 2.0 standard errors of the weighted one. The rejection does not say which number is wanted or why they differ, and at least four stories fit: the model may be misspecified, with heterogeneous contrasts that the weights average differently; sampling may be endogenous; a few heavily weighted records may have leverage; or the estimators may target different estimands, the sample's projection and the population's, both legitimate.

Which groups move the gap

In the divorce-law example below, one state mattered. If a single state, occupation, or education group accounted for the gap here, dropping it would close the gap. Figure 4.3 drops each in turn and re-estimates both regressions. The weighted coefficient is the smaller one, so the gap, WLS minus OLS, is negative. The largest single move is 0.006 log points, and it widens the gap: without BA+ workers, the gap becomes −0.010. The state that matters most is California, whose removal gives −0.006; the occupation group that matters most is Education & library (−0.002). The gap is spread thinly across many groups, which fits the heterogeneity reading better than a single sampling anomaly.

No single state, occupation, or education group explains the small WLS–OLS gap

WLS minus OLS adjusted coefficient after dropping one group at a time (largest movers shown)

Show data

Group	Estimate for	Estimate
State	Drop California	−0.006
State	Drop Georgia	−0.004
State	Drop Texas	−0.005
State	Drop North Carolina	−0.004
State	Drop Minnesota	−0.005
Occupation group	Drop Education & library	−0.002
Occupation group	Drop Office & admin support	−0.003
Occupation group	Drop Healthcare practitioners	−0.006
Occupation group	Drop Architecture & engineering	−0.005
Occupation group	Drop Management	−0.005
Education	Drop BA+	−0.010
Education	Drop Some college	−0.002
Education	Drop Less than HS	−0.004
Education	Drop HS	−0.004

Figure 4.3.Dropping any one state, occupation group, or education level leaves the weighted–unweighted gap small.Note: Headline specification. Rows are the five states and five occupation groups whose removal changes the gap most, plus all four education levels.Source: IPUMS USA, ACS 2024 1-year (SAMPLE 202401), analysis/usa/acs/part_2020_2024.parquet; replicate weights REPWTP1–80 from analysis/usa/acs_repwt/part_2020_2024.parquet

Where they do not agree

Predict first

Split the sample by education and estimate the regression separately in each of the four groups. Will OLS and WLS agree as closely in every group as they do in the whole?

The headline agreement is an average, and an average can hide a disagreement in one of its parts. Figure 4.4 estimates the regression separately within each education level, so that each group gets its own age and sex slopes, and compares OLS with WLS group by group. The standard error of each gap comes from the replicates: every one of the 80 replicate weights re-solves both regressions, so the gap has a sampling distribution of its own, and the comparison is a test of the gap rather than a glance at two overlapping intervals. In three of the four groups the two coefficients differ by less than one standard error of their gap (the largest is 0.8). Among workers whose highest attainment is some college, meaning some college or an associate degree, 266,367 records with a work-from-home share of 11.2%, the unweighted coefficient is 0.209 and the weighted one 0.196, a gap of −0.013 (95% CI −0.019–−0.007) log points, 4.6 standard errors of the gap and 6% of the coefficient. In dollars the same group gives $8.12 an hour unweighted and $7.36 weighted, 9% of the coefficient at 6.4 standard errors.

Four separate regressions, one per education level: where weighting matters

Table of the work-from-home coefficient estimated separately within each of four education levels, unweighted and PERWT-weighted, in logs and in dollars; the two estimates agree within a standard error in three groups and differ by several standard errors of their gap in the some-college group.
Education	Records	WFH share	OLS, log points	WLS, log points	WLS − OLS	Gap ÷ its SE	OLS, $/hour	WLS, $/hour	Gap ÷ its SE, levels
Less than HS	53,748	4.4%	0.118	0.117	−0.001	−0.1	$4.61	$4.30	−0.9
HS	204,437	6.4%	0.170	0.167	−0.004	−0.8	$5.60	$5.39	−1.1
Some college	266,367	11.2%	0.209	0.196	−0.013	−4.6	$8.12	$7.36	−6.4
BA+	459,923	20.0%	0.216	0.215	−0.001	−0.4	$9.73	$9.52	−1.9

Figure 4.4.Within three education groups the weighted and unweighted coefficients agree; within the some-college group they differ by several standard errors of their gap, and by more in dollars than in logs.Note: Each row is a regression of the outcome on WFH, age, age², and sex within one education level, so the age–sex slopes differ by group. Gap SE: 80 SDR replicates, each replicate re-solving both regressions (the OLS side applies the replicate perturbation to equal case weights). WFH shares are PERWT-weighted.Source: IPUMS USA, ACS 2024 1-year (SAMPLE 202401), analysis/usa/acs/part_2020_2024.parquet; replicate weights REPWTP1–80 from analysis/usa/acs_repwt/part_2020_2024.parquet

What sets this group apart is not its weights, whose coefficient of variation, 0.91, is close to the whole sample's 0.89. It is that inside the group the home–commute contrast varies with the characteristics the weights track, so the records the weights push up carry smaller contrasts: the DuMouchel–Duncan interaction from above, concentrated in one place. The two regressions average the same contrasts in different mixtures, and the population mixture is the one PERWT delivers. For this group the referee's question is not about the second decimal. It is the question of which population the coefficient describes, and the answer moves the coefficient by 6% in logs and 9% in dollars. Two lessons carry back to the headline. Agreement in the aggregate is not a license to skip the check in the parts a paper will report separately. And a levels specification exposes what a log specification smooths, which is the pattern Solon, Haider, and Wooldridge found in their divorce regressions.

The standard error: which repetition?

Predict first

Figure 4.5 lists five standard errors for the same coefficient. Which do you expect to be largest, and by roughly how much?

Agreement of point estimates says nothing about uncertainty. A standard error describes a repetition, and the five in Figure 4.5 describe different ones.

Same coefficient, five standard errors

Table of standard errors for the adjusted work-from-home coefficient under five variance estimators. Most agree closely; clustering by state is several times larger because it answers a different question.
Variance estimator	OLS	WLS	What it imagines repeating
Classical (iid)	0.0018	0.0018	independent draws of workers with a common error variance
Heteroskedasticity-robust (HC1)	0.0019	0.0024	independent draws of workers
Clustered by household (CR1)	0.0019	0.0024	independent draws of households
ACS replicate weights (SDR, 80)	0.0019	0.0023	the ACS address sample and its weighting, as the Census Bureau encodes them
Clustered by state (CR1, 51 clusters)	0.0096	0.0088	independent draws of states, which the ACS does not do

Figure 4.5.The robust, household-clustered, and replicate standard errors nearly coincide; the state-clustered one is several times larger because it imagines a repetition the ACS never performs.Note: Headline specification (WFH, age, age², sex, education). The OLS entry in the replicate row applies each replicate's perturbation (REPWTP_r ÷ PERWT) to equal case weights, an approximation because the replicates also re-run the weighting adjustments.Source: IPUMS USA, ACS 2024 1-year (SAMPLE 202401), analysis/usa/acs/part_2020_2024.parquet; replicate weights REPWTP1–80 from analysis/usa/acs_repwt/part_2020_2024.parquet

The classical WLS standard error (0.0018) treats PERWT as a precision weight and is too small. The heteroskedasticity-robust (0.0024) and household-clustered (0.0024) standard errors are larger. The design-based answer uses the ACS's 80 replicate weights, built by successive difference replication (SDR): re-solve the weighted regression once with each replicate weight $w_i^{(r)}$ and combine the deviations as $\Var(\hat\beta) = \tfrac{4}{80}\sum_{r=1}^{80}(\hat\beta_r - \hat\beta)^2$ with 79 degrees of freedom (In code). The Census Bureau built each replicate weight by rerunning its weighting on perturbed base weights, so the SDR standard error reflects the sample design and the weight construction together. It is 0.0023, 0.94 times the household-clustered value: the ACS samples addresses, so households (702,163 of them here) are its only clusters, and stratification and calibration trim a little off the variance. Applying the same replicate perturbations to equal case weights gives an approximate design-based standard error for OLS, and the WLS variance is 1.37 times as large: under this design, as under the robust formulas, the unweighted estimator is the more precise one.

Clustering by state (51 clusters, counting the District of Columbia) gives 0.0088, 3.9 times the replicate standard error. The arithmetic is right; the question is different. It imagines a repetition that redraws whole states, and neither the ACS sample nor anything in this descriptive target is drawn or assigned by state. Chapter 6 sets out when such a repetition is the relevant one.

The Moulton factor when the regressor varies within clusters

Chapter 3 used the factor of Moulton (1990) for a regressor that is constant within clusters. When the regressor varies within clusters and cluster sizes $m$ vary, the factor becomes

1 + \Big[\tfrac{\Var(m)}{\bar m} + \bar m - 1\Big]\rho_x\,\rho_u

where $\bar m$ is the mean cluster size and $\rho_x$ and $\rho_u$ are the within-cluster correlations of the regressor and the error. It approximates how much clustering inflates the variance of an OLS coefficient relative to the classical formula. It is a special-case regression analogue of the clustering component of a design effect, not the design effect itself (Appendix B): it prices clustering only, and it assumes that every pair of errors in a cluster is equally correlated. The observed inflation is the ratio of the cluster-robust to the classical variance of the unweighted coefficient. For households the factor predicts 1.04 against an observed 1.09. For states it predicts 6.86 against an observed 27.64. If the home–commute contrast differs from state to state, two home workers in the same state share more of their error than a home worker and a commuter do, so errors are not equally correlated and the factor understates the inflation; with clusters as large as states, the size term turns a modest departure into a large miss.

Box

Provenance: the divorce-law regressions

The lead example in Solon, Haider, and Wooldridge is a panel of state divorce rates from 1956 to 1988, used by Friedberg (1998) and by Wolfers (2006) to estimate the effect of unilateral divorce laws and re-estimated by Lee and Solon (2011). Friedberg and Wolfers weighted each state-year by population to correct heteroskedasticity. In the log specification of their Table 1, the robust standard errors for OLS over the first eight years after reform are about half those for WLS: weighting by population made the estimates less precise, which is Dickens's point. In levels, OLS and WLS differed sharply, and WLS without California moved toward OLS, which the authors read as a sign of heterogeneous effects rather than a verdict for either estimator. This box is provenance only: it reproduces none of those estimates, and the pattern has been checked against the working-paper version of the article, not the published tables.

In code

The whole chapter is one weighted least-squares solve, repeated. The replicate standard error re-solves it once per replicate weight and combines the deviations with the SDR constant:

python

import numpy as np
 
def wls(X, y, w):
    Xw = X * w[:, None]
    return np.linalg.solve(Xw.T @ X, Xw.T @ y)
 
beta = wls(X, y, perwt)                          # X: intercept, D, age, age^2, female, education
reps = np.array([wls(X, y, repwt[:, r]) for r in range(80)])
se = np.sqrt(4 / 80 * ((reps - beta) ** 2).sum(axis=0))   # t with 79 df

The OLS column of every figure is the same call with w equal to one. The DuMouchel–Duncan test is an OLS fit with X, the normalized weight, and their products as regressors, followed by a joint test of the product terms with household-clustered standard errors. The gap test of Figure 4.4 is the same loop run twice per replicate, once with repwt[:, r] and once with repwt[:, r] / perwt applied to equal case weights, and the SDR constant applied to the replicate differences.

Note

A new case

A referee sees that your weighted and unweighted coefficients agree to the second decimal and concludes that the weights do not matter and that the coefficient is the population's average effect. Which parts of that conclusion does the agreement support, and what has it not tested?

One answer

It supports only the first part, and only for this coefficient in this sample: the weights do not reshuffle the contrasts much here. It does not show that the model is correctly specified, that either coefficient is the population's average contrast (both are weighted averages of heterogeneous contrasts, with weights the regression chose), that inclusion is unrelated to the outcome in ways the weights do not capture, that the agreement survives in the subgroups the paper reports, or anything causal. It is also silent about the standard error, which depends on the repetition the analysis claims to describe, not on whether two point estimates agree.

What we learned about the question

In 2024, full-time wage and salary workers who usually worked from home earned 0.203 log points more per hour than commuters of the same age, sex, and education, about 23% as a proportional difference in geometric-mean wages, and 0.078 log points more, about 8%, within the same detailed occupation, industry, and state. Most of the raw gap corresponds to differences in who works from home and in which jobs. Within the some-college group the weighted contrast is 0.196 log points against an unweighted 0.209, the one place in the chapter where the referee's question changes the number.

What the method changed

Weighting moved the adjusted coefficient by 0.005 log points; the choice of controls moved it far more. Split by education, weighting moved one group's coefficient by 6% in logs and 9% in dollars and left the other three alone. Weighting did cost precision. The standard error came from the 80 replicate weights, which repeat the sampling and the weighting, and a state-clustered standard error answered a question about a repetition the survey does not perform.

What this analysis does not establish

A causal effect of working from home. Selection into jobs that allow it, and into working from home within those jobs, is not addressed, and the occupation-adjusted estimate conditions on job choices made alongside that decision.
A telework contrast. TRANWORK records the primary means of getting to work, so hybrid workers who mostly commute count as commuters. A telework measure such as the CPS's defines a different contrast that no weighting reconciles.
Which coefficient is right. Agreement between OLS and WLS does not show that either one is the population's average contrast, and the DuMouchel–Duncan rejection does not show that either one is wrong. In the some-college group the two estimate different projections, the sample's and the population's; the weighted one describes the population, and neither is a causal effect.
Why the some-college group differs. The chapter shows that heavily weighted workers in that group have smaller contrasts; it does not identify which characteristic the weights are tracking.
Error-free wages. Hourly wages divide annual income by usual hours and the midpoint of a weeks-worked band, with state top codes as released; neither approximation is corrected.