Part II Chapter 5

Hidden Weights Under Heterogeneous Effects

An adjusted regression averages heterogeneous contrasts with weights nobody chose. Here is how to see them.

DataACS 2024 · implicit regression weights

Whose wage gap is it?

You ran the regression with the controls your field expects, and you are about to describe the coefficient as the average effect of the treatment, holding the controls fixed. Most readers will take it that way. But when the effect differs across the groups the controls define, and it usually does, the coefficient is a weighted average of the group contrasts with weights the regression chose and nobody reports. Survey weights do not choose them and cannot correct them. This chapter makes the weights visible in a case where they move the answer, and in a second case where they are just as unequal and do not.

A report on the 2024 American Community Survey wants one sentence: among full-time wage and salary workers, how much more do people who work from home earn than people who commute, holding occupation and education fixed? Chapter 4 supplied the regression. Refine its education dummies into 85 occupation-by-education cells, keep the survey weights, and the coefficient is 0.118 (95% CI 0.114–0.123) log points. The same 85 cells each have their own home–commute contrast, and those contrasts can be averaged by hand, each cell counted by its share of workers. That average is 0.088 (95% CI 0.082–0.095). Same workers, same PERWT, same cell contrasts, and the regression's number is 1.35 times the population's.

Both numbers are defensible. The difference between them is a choice of weights, and nobody made it. Written out, a regression with a full set of cell effects is a weighted average of the cell contrasts in which each cell counts by its population times the variance of the work-from-home indicator inside it (Angrist 1998). A cell where home workers and commuters are both plentiful counts heavily. A cell where almost everyone commutes barely counts, however many workers it holds. College-educated managers, 8.6% of the workforce, receive 14.7% of the regression's weight. Transportation and moving workers with a high school diploma, 3.1% of the workforce, receive 0.8%. The ten most-weighted cells hold 44% of workers and 62% of the weight. Seen from the other side, in the decomposition of Słoczyński (2022), home workers are 13.7% of the workforce and the term for their contrast carries 83% of the coefficient.

A reader should care for two reasons. The contrasts differ across cells, from −0.321 to 0.327 log points, and they tend to be larger where home work is common, so the cells the regression favors are the cells with the larger gaps. And the survey weights cannot fix this. They make each cell's population and home-work share estimates for the nation; they do not decide how the cells are combined. Chapter 4's coefficient, 0.203, agreed with its unweighted twin, and the cell regression's unweighted twin, 0.122, agrees with it as well. The weights that separate the regression from the population average are not survey weights. This chapter makes them visible and says what to report instead.

Who works from home

Working from home (WFH) means, as in Chapter 4, usually working at home in the survey's reference week, and it is concentrated in particular jobs. Across the 23 major occupation groups of the 2018 Standard Occupational Classification, the share ranges from 3.1% in Food preparation & serving to 39.2% in Computer & math (Figure 5.1), and it rises steeply with education.

Working from home is concentrated in a few occupation groups

Share of full-time wage and salary workers aged 25–64 whose main way to work is working at home, 2024

Show data

Category	Value	95% CI
Computer & math	39.2%	38.6%–39.8%
Business & financial	29.7%	29.3%–30.1%
Arts, design, media & sports	25.1%	24.2%–26.0%
Legal	20.9%	20.0%–21.8%
Management	19.6%	19.3%–19.8%
Office & admin support	16.3%	16.0%–16.6%
Sales	15.4%	15.0%–15.7%
Architecture & engineering	15.1%	14.6%–15.6%
Life, physical & social science	15.0%	14.3%–15.7%
Community & social service	11.9%	11.3%–12.5%
Healthcare support	8.9%	8.4%–9.3%
Healthcare practitioners	7.7%	7.4%–8.0%
Personal care & service	6.7%	6.1%–7.2%
Education & library	6.1%	5.9%–6.3%
Military specific	5.7%	4.6%–6.9%
Protective service	4.9%	4.6%–5.3%
Farming, fishing & forestry	4.1%	3.3%–4.9%
Production	3.7%	3.5%–3.9%
Installation & repair	3.7%	3.4%–4.0%
Transportation & moving	3.5%	3.3%–3.8%
Building & grounds	3.5%	3.2%–3.9%
Construction & extraction	3.5%	3.2%–3.8%
Food preparation & serving	3.1%	2.9%–3.3%

Figure 5.1.Working from home is concentrated in a few occupation groups.Note: Intervals: 80-replicate SDR, t with 79 df. Occupation groups are 2018 SOC major groups built from OCC code ranges.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 contrasts vary too. Within education groups, the home–commute gap is 0.217 log points for college graduates and 0.119 for workers without a diploma. When both the chance of working from home and the wage contrast differ across groups, any single coefficient is a weighted average of group contrasts, and the question is who chose the weights.

Cells and their contrasts

The cells cross the 23 occupation groups with four education levels. A contrast needs both home workers and commuters in the cell, so 7 thin cells were merged into a neighbor, leaving 85; no record is dropped. The model is Chapter 4's with the education dummies refined into cells:

y_i = \beta D_i + \alpha_{c(i)} + z_i'\gamma + \varepsilon_i ,

where $y_i$ is log hourly wage, $D_i$ indicates working from home, $c(i)$ is person $i$ 's cell, and $z_i$ holds age, age squared, and a female indicator. The cell contrast $\hat\tau_c$ is the survey-weighted difference in mean log wage between home workers and commuters in cell $c$ , after subtracting the fitted age–sex profile $z_i'\hat\gamma$ . That profile is common to all cells and fitted on the whole sample, so every contrast in this chapter is net of it. The contrasts run from −0.321 to 0.327, and the share working from home from 2.4% to 41.0%.

How the cells were built

The base partition has 92 cells. Within each occupation group, a thin education cell is merged into its neighbor until every cell has at least 100 records and at least 30 workers on each side of the home–commute divide. All 7 merges were "less than high school" cells in occupations where few workers lack a diploma, folded into the high school cell. Because the age–sex profile is fitted once on the whole sample, the cell contrasts are not the contrasts that separate regressions within each cell would give.

Which cells the regression listens to

Predict first

The survey-weighted regression with cell fixed effects is an exact weighted average of the 85 cell contrasts. Before reading on: do the largest cells get the most weight? What happens to a large cell in which almost nobody works from home?

A comparison within a cell needs both kinds of worker. A cell where almost everyone commutes offers only a handful of home workers, so however large it is, it says little about the home–commute contrast, and a regression that compares only within cells leans on the cells where both groups are plentiful. With a full set of cell fixed effects, the coefficient equals

\hat\beta = \frac{\sum_c \omega_c\, \hat\tau_c}{\sum_c \omega_c}, \qquad \omega_c = N_c\, p_c (1 - p_c),

where $N_c$ is the cell's weighted population and $p_c$ its weighted share working from home (Angrist 1998). A cell's weight is its size times the variance of the work-from-home indicator inside it, so the answer to the prediction is no: a cell where almost everyone commutes has $p_c(1-p_c)$ near zero, however large it is. Aronow and Samii (2016) write the same weights at the level of individuals, $\omega_i = (D_i - \hat E[D_i \mid X_i])^2$ ; with cell effects alone, the individual weights times their survey weights $w_i$ add up to $N_c p_c(1-p_c)$ .

The weights are never negative in this form, and survey weights enter them only through $N_c$ and $p_c$ , making those two quantities estimates for the population rather than the sample. Survey weighting does not decide that the factor $p_c(1-p_c)$ belongs in the average; the regression does.

Why the identity is exact with age and sex in the model

The age–sex coefficients $\hat\gamma$ come from the same least-squares fit as $\hat\beta$ , so holding them at $\hat\gamma$ and refitting only the treatment and cell effects returns the same $\hat\beta$ . That refit is a regression of the adjusted outcome $y_i - z_i'\hat\gamma$ on the treatment and the cell dummies alone, to which Angrist's algebra applies. The build recomputes the coefficient from the adjusted cell contrasts and weights and matches the regression to machine precision; Appendix B gives the derivation, including this step.

Cells that dominate the regression, and large cells that barely count

Table of the cells with the largest implicit weight and the largest population shares, with their WFH shares and contrasts; large low-WFH cells such as production and transportation carry little regression weight.
Cell	Records	Share of workers	WFH share	Regression weight	Contrast (log points)	SE
Management · BA+	89,147	8.6%	24.1%	14.7%	0.184	0.007
Business & financial · BA+	53,082	5.2%	32.1%	10.6%	0.092	0.005
Computer & math · BA+	38,680	3.9%	41.0%	8.8%	0.045	0.006
Office & admin support · some college	43,926	4.4%	15.8%	5.5%	0.070	0.009
Office & admin support · BA+	31,511	3.1%	21.0%	4.8%	0.171	0.011
Sales · BA+	26,324	2.7%	25.4%	4.7%	0.327	0.013
Management · some college	31,206	3.1%	14.0%	3.5%	0.236	0.011
Healthcare practitioners · BA+	49,231	4.8%	7.7%	3.2%	−0.011	0.012
Education & library · BA+	56,654	5.2%	6.3%	2.9%	0.075	0.011
Transportation & moving · HS	28,064	3.1%	2.9%	0.8%	−0.022	0.029
Transportation & moving · some college	18,219	2.0%	3.7%	0.7%	−0.020	0.037
Production · HS	24,272	2.5%	2.8%	0.6%	0.041	0.033

Figure 5.2.Regression weight goes to cells where many work from home; large cells where few do barely count.Note: The eight cells with the most implicit weight, the six largest cells, and the three largest cells where fewer than 5% work from home. SE: 80-replicate SDR.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

Figure 5.2 lists the cells with the most weight beside the largest cells. The Management · BA+ cell, where 24% work from home, holds 8.6% of workers and 14.7% of the weight. The largest cell in which almost nobody works from home, Transportation & moving · HS (2.9% work from home), holds 3.1% of workers and 0.8% of the weight. The weights matter because the contrasts differ. College-educated healthcare practitioners are 4.8% of workers, with 8% working from home and a contrast of −0.011; they get 3.2% of the weight. College-educated sales workers are a smaller group, 2.7% of workers, but 25% of them work from home and their contrast is 0.327; they get 4.7%.

One set of contrasts, several averages

Figure 5.3 sets the two cell regressions beside three averages of the same 85 contrasts: the population-share average, which weights each cell by its share of workers; the home-worker-share average, by its share of home workers; and the commuter-share average, by its share of commuters. Chapter 4's coefficient appears as a muted reference. The three averages copy the weighting schemes of the population average treatment effect, the effect on the treated, and the effect on the untreated, so the figure also labels them PATE-, ATT-, and ATU-analogues. The resemblance is in the weighting only; nothing here identifies a causal effect.

One set of cell contrasts, several defensible averages

WFH − commuter difference in log hourly wage, 85 occupation-by-education cells, 2024 ACS

Unweighted
Weighted
Treated
Control

Show data

Group	Estimate for	Estimate	95% CI	n
Regression coefficients	Unweighted regression, cell fixed effects	0.122	0.118–0.125	984,475
Regression coefficients	Survey-weighted projection, cell fixed effects	0.118	0.114–0.123	984,475
Regression coefficients	Chapter 4 headline (education only, weighted)	0.203	0.199–0.208	984,475
Averages of the same cell contrasts	Population-weighted (PATE-analogue)	0.088	0.082–0.095	984,475
Averages of the same cell contrasts	WFH-weighted (ATT-analogue)	0.120	0.115–0.124	984,475
Averages of the same cell contrasts	Commuter-weighted (ATU-analogue)	0.083	0.076–0.090	984,475

Figure 5.3.The survey-weighted regression lands next to the home-worker-share average, not the population-share average.Note: Weighted rows: 80-replicate SDR, t with 79 df. OLS intervals use household-clustered (CR1) standard errors and the normal critical value. The projection equals the average of the cell contrasts with weights N_c p_c(1 − p_c).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 population-share average is 0.088. The home-worker-share average is 0.120 and the commuter-share average 0.083, and the first exceeds the second by 0.037 (95% CI 0.032–0.042) log points: the contrast tends to be larger in cells where working from home is common. The survey-weighted regression, 0.118, sits beside the home-worker-share average, 0.030 (95% CI 0.027–0.034) above the population-share average. The unweighted regression gives 0.122.

The ledger holds every field fixed but one, the implicit weights, which differ across the four summaries.

Analysis Ledger

Whose wage gap? Averaging the same cell contrasts in different ways

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: Descriptive averages of occupation-by-education cell contrasts in log hourly wage (WFH minus commuters, net of a common age–sex profile): population-weighted (PATE-analogue), WFH-weighted (ATT-analogue), commuter-weighted (ATU-analogue), and the survey-weighted projection coefficient.
Estimator: Weighted cell moments from PERWT; contrasts are within-cell differences in mean log wage minus the fitted age–sex profile; the projection is WLS with cell fixed effects.
Explicit weights: PERWT for every weighted quantity; equal case weights for the OLS comparison.
Implicit weights: Projection: N_c p_c(1 − p_c). PATE-analogue: N_c. ATT-analogue: N_c p_c. ATU-analogue: N_c(1 − p_c).
Source of randomness: The ACS sample and response process; no treatment assignment.
Variance estimator: SDR, 80 replicates; every contrast, cell weight, and average is recomputed per replicate; df = 79.
Assumptions: 7 small cells merged within occupation (n ≥ 100, 30+ in each arm); common age–sex profile across cells; descriptive, not causal; TRANWORK construct caveat applies.
Numbers reported: 0.118log points
0.088log points
0.120log points
0.083log points
0.122log points

The unweighted regression gives every record the same case weight, but it does not give every cell's contrast the same weight, or a weight in proportion to its sample size: its implicit weights are $n_c \hat p_c(1-\hat p_c)$ . Across cells they are almost identical to the survey-weighted regression's (correlation 1.00). Equal case weights do not imply equal weighting of heterogeneous conditional effects.

The regression-effective sample

A survey-weighted regression can represent the population correctly in the design sense, every cell at its estimated population size, while placing highly unequal implicit weight on heterogeneous conditional contrasts. The implicit weights form the third of the booklet's four effective samples (Appendix B), a composition rather than a count. Figure 5.4 sums them within occupation groups and sets each group's share of the weight against its share of workers.

The regression's effective sample is not the population

Share of workers vs share of the survey-weighted regression's implicit weight, by occupation group

Show data

Row	Share of workers	Share of regression weight	Change
Management	13.7%	19.7%	+6.0 pp
Business & financial	7.4%	14.4%	+7.0 pp
Office & admin support	10.6%	13.4%	+2.8 pp
Computer & math	5.2%	11.6%	+6.4 pp
Sales	7.2%	8.3%	+1.1 pp
Healthcare practitioners	7.3%	4.8%	−2.4 pp
Architecture & engineering	3.0%	3.5%	+0.6 pp
Education & library	6.4%	3.4%	−3.0 pp
Arts, design, media & sports	1.6%	2.8%	+1.2 pp
Transportation & moving	6.8%	2.2%	−4.6 pp
Healthcare support	2.8%	2.1%	−0.7 pp
Community & social service	2.0%	2.0%	0.0 pp
Legal	1.2%	1.9%	+0.7 pp
Production	5.7%	1.9%	−3.8 pp
Life, physical & social science	1.4%	1.6%	+0.3 pp
Construction & extraction	4.3%	1.4%	−3.0 pp
Installation & repair	3.3%	1.1%	−2.2 pp
Protective service	2.4%	1.1%	−1.4 pp
Food preparation & serving	3.1%	0.9%	−2.2 pp
Building & grounds	2.5%	0.8%	−1.7 pp
Personal care & service	1.2%	0.7%	−0.5 pp
Military specific	0.4%	0.2%	−0.2 pp
Farming, fishing & forestry	0.4%	0.2%	−0.3 pp

Figure 5.4.Occupations where home work is common gain regression weight; large occupations where it is rare lose most of theirs.Note: Implicit weight of a cell = N_c p_c (1 − p_c) (Angrist 1998), summed within occupation group.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 concentration in the opening runs through the whole distribution. The 26 cells where fewer than one worker in twenty works from home hold 26% of workers and 8% of the weight; cells where a quarter or more work from home hold 16% of workers and 32% of the weight. Measured as an effective number of equally weighted cells (the inverse of the sum of squared shares), the population's 34 shrink to 18 in the regression, and a worker in the most balanced cell counts 10 times as much as a worker in the least balanced one.

In code: the four averages from the cell table

The implicit weights and the three averages can be reproduced from the cell data behind the interactive below.

python

import numpy as np
 
share, p, tau = cells["pop_share"], cells["treat_share"], cells["effect"]
omega = share * p * (1 - p)                     # Angrist (1998) implicit weights
 
projection = np.sum(omega * tau) / np.sum(omega)    # = survey-weighted coefficient
pate = np.sum(share * tau) / np.sum(share)           # population-share average
att = np.sum(share * p * tau) / np.sum(share * p)    # home-worker-share average
atu = np.sum(share * (1 - p) * tau) / np.sum(share * (1 - p))  # commuter-share average

Every aggregate in this chapter, including the population-share average, gets its standard error by rerunning the whole calculation on each of the 80 replicate weights.

The same cells, a second contrast: the gender gap

Predict first

Keep the 85 cells and change the contrast to women against men. Women are 46.1% of these workers, 2.2% of the Installation & repair · less than HS cell and 88.4% of the Legal · some college cell, so the regression will again give the lopsided cells little weight. Will its coefficient miss the population-share average by as much as it did for working from home?

The gender wage gap within occupations is a standard object in labor economics and sociology, and the usual estimate of it is the coefficient on a female indicator in a regression with occupation controls (Blau and Kahn 2017). The model is the one above with the roles exchanged: the female indicator is the treatment, working from home joins age and age squared among the controls, and the cells are unchanged, since every cell holds at least 30 women and 30 men. The cell contrasts run from −0.474 to 0.073 log points and the female share from 2.2% to 88.4%.

The implicit weights are as unequal as before. The 23 cells in which women are fewer than a fifth or more than four fifths of workers hold 20.1% of the workforce and receive 10.5% of the regression's weight; the ten most-weighted cells hold 43.3% of workers and 49.5% of the weight; the effective number of cells falls from 33.8 to 27.5. Yet the survey-weighted regression gives −0.198 (95% CI −0.201–−0.195) and the population-share average −0.198 (95% CI −0.201–−0.194), a difference of −0.001 (95% CI −0.002–0.001) (Figure 5.5); the unweighted regression gives −0.206.

The gender gap: unequal implicit weights, and the same answer

Female − male difference in log hourly wage, the same 85 occupation-by-education cells, 2024 ACS

Unweighted
Weighted
Design-based

Show data

Group	Estimate for	Estimate	95% CI	n
Regression coefficients	Unweighted regression, cell fixed effects	−0.206	−0.209–−0.204	984,475
Regression coefficients	Survey-weighted projection, cell fixed effects	−0.198	−0.201–−0.195	984,475
Averages of the same cell contrasts	Population-share average (PATE-analogue)	−0.198	−0.201–−0.194	984,475
Averages of the same cell contrasts	Women-share average (ATT-analogue)	−0.185	−0.189–−0.182	984,475
Averages of the same cell contrasts	Men-share average (ATU-analogue)	−0.208	−0.213–−0.203	984,475

Figure 5.5.For the gender gap the survey-weighted regression and the population-share average coincide, although the implicit weights are as unequal as in the work-from-home case.Note: Weighted rows: 80-replicate SDR, t with 79 df. OLS intervals use household-clustered (CR1) standard errors and the normal critical value. Contrasts are net of a common age and work-from-home profile.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 reason is in Figure 5.6. The regression drifts from the population average only when the contrasts covary with the overlap term $p_c(1-p_c)$ , and here they barely do: the population-share-weighted correlation between them is −0.02. The gap in the lopsided cells, −0.176 log points, is close to the gap in the mixed cells, −0.203, and reweighting cells whose contrasts are alike changes nothing. In the work-from-home case the cells that gained weight were the cells with the larger contrasts, and the coefficient followed them.

Where the gender regression's weight goes

Cells grouped by their female share; contrasts are population-share averages within the band

Table of three bands of cells by female share, showing the number of cells, their share of workers, their share of the regression's implicit weight, and their average gender contrast; mixed cells gain weight and segregated cells lose it, while the average contrast differs little between the first two bands.
Female share of the cell	Cells	Share of workers	Regression weight	Average contrast (log points)
Under 20% women	16	15.1%	6.9%	−0.197
20–80% women	62	79.9%	89.5%	−0.203
Over 80% women	7	5.0%	3.6%	−0.114

Figure 5.6.Mixed cells gain regression weight and lopsided cells lose it, but the average contrast in the two groups is nearly the same.Note: Implicit weight of a cell = N_c p_c (1 − p_c) with p_c the cell's PERWT-weighted female share.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

Two smaller differences follow the same logic. The women-share average is −0.185 and the men-share average −0.208, 0.023 (95% CI 0.018–0.028) apart, because the gap is smaller in the few cells where women predominate. The regression sits between the two rather than beside the women-share average: with a treatment that splits the population nearly in half, the weights $p_c(1-p_c)$ tilt toward neither group. And Słoczyński's weight on the women's term is 60% against a female share of 46.1%, the same over-weighting of the smaller group as before, in a case where it happens not to matter. Unequal implicit weights are necessary for the regression to miss the population average; they are not sufficient. The diagnostic is not the spread of the weights but their covariance with the contrasts, and Figure 5.6 is the table to report.

Analysis Ledger

The within-cell gender wage gap: the same averages, a second outcome

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: Descriptive averages of occupation-by-education cell contrasts in log hourly wage (women minus men, net of a common age and work-from-home profile): population-weighted (PATE-analogue), women-weighted (ATT-analogue), men-weighted (ATU-analogue), and the survey-weighted projection coefficient.
Estimator: Weighted cell moments from PERWT; contrasts are within-cell differences in mean log wage minus the fitted age and work-from-home profile; the projection is WLS with cell fixed effects.
Explicit weights: PERWT for every weighted quantity; equal case weights for the OLS comparison.
Implicit weights: Projection: N_c p_c(1 − p_c) with p_c the female share. PATE-analogue: N_c. ATT-analogue: N_c p_c. ATU-analogue: N_c(1 − p_c).
Source of randomness: The ACS sample and response process; sex is not assigned.
Variance estimator: SDR, 80 replicates; every contrast, cell weight, and average is recomputed per replicate; df = 79.
Assumptions: The chapter's 85 cells reused unchanged (every cell has at least 30 women and 30 men); common age and work-from-home profile across cells; descriptive, not causal.
Numbers reported: −0.198log points
−0.198log points
−0.185log points
−0.208log points
−0.206log points

The variance-weighting machine

Predict first

In the panel below, make group 1 small but with a treated share near one half, and group 0 large with few treated. Before moving the effect sliders: will the regression land near the population average or near group 1's effect?

The top of the panel is the two-group version of the argument: a regression of the outcome on the treatment and a group dummy. Set group 1's share of the population, each group's treated share, and each group's effect. The panel computes the PATE, the ATT, the ATU, and the probability limit of the regression coefficient, in which each group's weight is its population share times $p_g(1-p_g)$ . The two knife-edge buttons show the only ways the regression lands on the PATE: equal effects, or equal treatment variance in the two groups. The lower half loads the 85 real cells with their shares, contrasts (bands of $\pm 1.96$ replicate standard errors), and implicit weights; sort by the ratio of implicit weight to population share to see which cells the regression over- and under-counts.

The variance-weighting machine

Two groups, a binary treatment, and a regression with a group dummy. Which average of the two group effects does the regression coefficient estimate?

Share of the population in group 1 (s)35%

Treated share in group 1 (p₁)30%

Treated share in group 0 (p₀)6%

Effect in group 1 (τ₁)0.20

Effect in group 0 (τ₀)0.04

plim β̂ = [s·p₁(1−p₁)·τ₁ + (1−s)·p₀(1−p₀)·τ₀] / [s·p₁(1−p₁) + (1−s)·p₀(1−p₀)] = [0.073 × 0.20 + 0.037 × 0.04] / 0.110 = 0.147

population share · implicit regression weight ω

Group 135%67%

Group 065%33%

Equal effects? No, τ₁ − τ₀ = +0.16.
Equal treatment variance? No, p₁(1−p₁) = 0.210 and p₀(1−p₀) = 0.056.

Seed 1One simulated sample of 5,000 from these settings.

OLS coefficient in the sample: 0.183 (robust SE 0.017), against a probability limit of 0.147 and a PATE of 0.096.

What is held fixed. Two groups; a binary treatment with a constant effect inside each group; a regression of the outcome on the treatment and a group dummy, with no other controls. The averages are population quantities; only the one-sample check involves sampling.

Occupation-by-education cells: who works from home, and the wage contrast in each

Log hourly wage, people who work from home versus commuters, cell by cell. Implicit weight ω ∝ population share × p(1 − p), the weight each cell's effect gets in a survey-weighted regression with a full set of cell dummies.

Cells sorted by omega; click a column header to sort.

Management · BA+	8.6%	24%	0.184 ± 0.013	14.7%	1.72×
Business & financial · BA+	5.2%	32%	0.092 ± 0.010	10.6%	2.04×
Computer & math · BA+	3.9%	41%	0.045 ± 0.012	8.8%	2.27×
Office & admin support · some college	4.4%	16%	0.070 ± 0.017	5.5%	1.25×
Office & admin support · BA+	3.1%	21%	0.171 ± 0.022	4.8%	1.55×
Sales · BA+	2.7%	25%	0.327 ± 0.025	4.7%	1.77×
Management · some college	3.1%	14%	0.236 ± 0.021	3.5%	1.13×
Healthcare practitioners · BA+	4.8%	8%	−0.011 ± 0.024	3.2%	0.67×
Education & library · BA+	5.2%	6%	0.075 ± 0.022	2.9%	0.56×
Office & admin support · HS	2.7%	13%	0.083 ± 0.022	2.8%	1.03×
Architecture & engineering · BA+	2.2%	17%	0.105 ± 0.022	2.8%	1.30×
Business & financial · some college	1.5%	27%	0.110 ± 0.025	2.7%	1.84×
Sales · some college	2.3%	12%	0.302 ± 0.033	2.3%	1.02×
Computer & math · some college	1.1%	36%	0.175 ± 0.026	2.3%	2.16×
Arts, design, media & sports · BA+	1.1%	28%	0.170 ± 0.029	2.2%	1.90×
Community & social service · BA+	1.6%	12%	0.031 ± 0.035	1.6%	1.02×
Legal · BA+	1.0%	22%	0.062 ± 0.039	1.6%	1.62×
Life, physical & social science · BA+	1.2%	16%	0.176 ± 0.030	1.5%	1.26×
Healthcare practitioners · some college	2.0%	8%	−0.003 ± 0.030	1.4%	0.66×
Management · HS	1.7%	9%	0.164 ± 0.044	1.3%	0.76×
Sales · HS	1.9%	7%	0.208 ± 0.052	1.1%	0.59×
Business & financial · HS	0.6%	19%	0.171 ± 0.036	0.9%	1.43×
Healthcare support · some college	1.3%	8%	−0.148 ± 0.053	0.8%	0.67×
Transportation & moving · HS	3.1%	3%	−0.022 ± 0.057	0.8%	0.26×
Transportation & moving · some college	2.0%	4%	−0.020 ± 0.073	0.7%	0.33×
Production · HS	2.5%	3%	0.041 ± 0.065	0.6%	0.26×
Healthcare support · HS	0.9%	8%	−0.139 ± 0.056	0.6%	0.67×
Construction & extraction · HS	1.8%	3%	0.059 ± 0.062	0.6%	0.32×
Production · some college	1.8%	4%	0.024 ± 0.058	0.6%	0.32×
Architecture & engineering · some college	0.5%	12%	0.199 ± 0.052	0.6%	1.01×
Computer & math · HS	0.3%	28%	0.133 ± 0.060	0.5%	1.90×
Protective service · BA+	0.8%	7%	0.132 ± 0.073	0.5%	0.62×
Production · BA+	0.6%	9%	0.266 ± 0.068	0.5%	0.77×
Arts, design, media & sports · some college	0.3%	20%	0.186 ± 0.071	0.5%	1.49×
Healthcare support · BA+	0.5%	12%	−0.079 ± 0.080	0.5%	0.98×
Transportation & moving · BA+	0.8%	7%	0.099 ± 0.078	0.5%	0.57×
Installation & repair · some college	1.3%	4%	0.043 ± 0.054	0.4%	0.33×
Installation & repair · HS	1.3%	3%	−0.043 ± 0.083	0.4%	0.29×
Construction & extraction · some college	1.2%	4%	−0.100 ± 0.087	0.4%	0.32×
Protective service · some college	1.0%	4%	0.039 ± 0.084	0.4%	0.35×
Education & library · some college	0.7%	5%	0.229 ± 0.066	0.3%	0.45×
Building & grounds · HS	1.1%	3%	0.042 ± 0.112	0.3%	0.30×
Food preparation & serving · HS	1.2%	3%	0.006 ± 0.083	0.3%	0.26×
Office & admin support · less than HS	0.4%	9%	0.110 ± 0.077	0.3%	0.75×
Food preparation & serving · some college	0.9%	3%	0.060 ± 0.092	0.3%	0.30×
Healthcare support · less than HS	0.2%	13%	−0.098 ± 0.082	0.3%	1.07×
Healthcare practitioners · HS	0.4%	7%	0.007 ± 0.084	0.3%	0.65×
Construction & extraction · less than HS	1.0%	3%	−0.012 ± 0.078	0.3%	0.25×
Personal care & service · some college	0.4%	7%	−0.010 ± 0.102	0.2%	0.59×
Transportation & moving · less than HS	1.0%	3%	−0.070 ± 0.094	0.2%	0.26×
Community & social service · some college	0.3%	10%	0.116 ± 0.076	0.2%	0.86×
Legal · some college	0.2%	16%	0.075 ± 0.065	0.2%	1.28×
Management · less than HS	0.3%	8%	0.219 ± 0.121	0.2%	0.68×
Production · less than HS	0.8%	3%	0.004 ± 0.087	0.2%	0.26×
Building & grounds · some college	0.6%	4%	0.033 ± 0.100	0.2%	0.36×
Personal care & service · HS	0.4%	6%	0.041 ± 0.133	0.2%	0.50×
Protective service · HS or less	0.6%	4%	0.102 ± 0.100	0.2%	0.34×
Personal care & service · BA+	0.3%	9%	−0.144 ± 0.162	0.2%	0.73×
Installation & repair · BA+	0.3%	7%	0.108 ± 0.092	0.2%	0.64×
Sales · less than HS	0.4%	6%	0.185 ± 0.137	0.2%	0.49×
Architecture & engineering · HS or less	0.3%	9%	0.187 ± 0.100	0.2%	0.74×
Education & library · HS	0.4%	5%	0.300 ± 0.118	0.2%	0.43×
Food preparation & serving · BA+	0.4%	5%	0.075 ± 0.104	0.2%	0.41×
Construction & extraction · BA+	0.3%	6%	0.133 ± 0.155	0.2%	0.52×
Building & grounds · less than HS	0.6%	3%	0.003 ± 0.116	0.2%	0.25×
Arts, design, media & sports · HS or less	0.1%	11%	0.199 ± 0.231	0.1%	0.93×
Community & social service · HS or less	0.2%	10%	0.064 ± 0.105	0.1%	0.85×
Food preparation & serving · less than HS	0.6%	2%	0.002 ± 0.132	0.1%	0.22×
Building & grounds · BA+	0.2%	6%	−0.094 ± 0.177	0.1%	0.53×
Life, physical & social science · some college	0.1%	10%	0.106 ± 0.111	0.1%	0.81×
Business & financial · less than HS	0.1%	14%	0.064 ± 0.158	0.1%	1.11×
Legal · HS or less	0.1%	15%	0.164 ± 0.138	0.1%	1.20×
Installation & repair · less than HS	0.3%	3%	−0.059 ± 0.147	0.1%	0.26×
Military specific · BA+	0.2%	5%	−0.043 ± 0.158	0.1%	0.46×
Military specific · some college	0.1%	6%	−0.321 ± 0.199	0.1%	0.49×
Computer & math · less than HS	0.0%	20%	0.072 ± 0.193	0.1%	1.49×
Life, physical & social science · HS or less	0.1%	8%	0.183 ± 0.187	0.1%	0.70×
Personal care & service · less than HS	0.1%	5%	−0.108 ± 0.217	0.1%	0.48×
Farming, fishing & forestry · less than HS	0.2%	3%	0.044 ± 0.177	0.0%	0.25×
Farming, fishing & forestry · HS	0.1%	3%	−0.039 ± 0.236	0.0%	0.31×
Farming, fishing & forestry · BA+	0.0%	11%	0.153 ± 0.328	0.0%	0.92×
Military specific · HS or less	0.1%	8%	0.087 ± 0.195	0.0%	0.67×
Healthcare practitioners · less than HS	0.1%	6%	0.310 ± 0.478	0.0%	0.52×
Farming, fishing & forestry · some college	0.1%	5%	−0.280 ± 0.309	0.0%	0.43×
Education & library · less than HS	0.1%	4%	0.082 ± 0.260	0.0%	0.39×

Effect intervals are ±1.96 SE where an SE is supplied.

These contrasts are descriptive. Each cell effect is an adjusted difference in log hourly wage between people who did and did not receive the treatment in that cell, not a causal effect; the four averages differ only in how they weight the same cell contrasts.

With the settings in the prediction, the regression lands far nearer group 1's effect than the population average does. Suppose group 1 is a fifth of the population with half its members treated, and one member of group 0 in twenty is treated. Group 1's implicit weight is its share times one-quarter, and group 0's is its share times a little under one-twentieth, so a fifth of the population gets more than half of the regression's weight.

What the survey weights did and did not do

The opening's two numbers are both averages of the same 85 contrasts. The survey-weighted regression, 0.118, weights each cell by $N_c p_c(1-p_c)$ ; the population-share average, 0.088, weights it by $N_c$ . PERWT did its job in both: every cell's population and share working from home are estimated for the population, and the weighted and unweighted regressions agree, as in Chapter 4. What the weights cannot do is choose the average. With heterogeneous contrasts, a regression that compares within cells leans on overlap cells, here the college-educated managers, business and computer workers, and office staff among whom home work is common enough to compare. The population's average contrast is a different and smaller number, and survey weights will not turn a regression coefficient into it (Solon, Haider, and Wooldridge 2015).

Four habits follow for anyone reporting a regression like this one.

Name the target. The population projection, a population-weighted average of conditional contrasts, and an average for the treated are three different numbers.
If the population average is the target, estimate it directly. Compute the contrast in each cell, or interact the treatment with the cells, and aggregate with population shares. This counts small cells, whose contrasts rest on few home workers, at full population weight, and it depends on how the workers were divided into cells.
Report the regression-effective sample next to the coefficient. The implicit weights by group against population shares, as in Figure 5.4 and the table inside the interactive, and the group contrasts beside them, since unequal weights matter only where the contrasts differ (Figure 5.6).
When the controls enter additively rather than as a full set of cells, check the two representations in the next section. Some implied weights can turn negative, and the weight on the treated group's term can sit far from the treated share.

Going further: signed weights and group weights

The headline needs nothing beyond the cell weights above. Two further representations of the same kind of coefficient apply when controls enter additively, and their labels invite confusion. Each weights something different.

Representation	What is weighted	Model	Can a weight be negative?
Cell weights $N_c\, p_c(1-p_c)$ (Angrist 1998)	cell contrasts	a full set of cell effects	no
Individual weights $(D_i - \hat E[D_i \mid X_i])^2$ (Aronow and Samii 2016)	individuals' conditional contrasts	linear controls, with a treatment probability linear in them	no
Signed weights proportional to $D_i - \hat\ell_i$ (Chattopadhyay and Zubizarreta 2023)	individual outcomes, treated against controls	any linear controls	yes
Group weights $\omega_1$ and $\omega_0$ (Słoczyński 2022)	the two groups' average partial linear effects	additive linear controls	no, but the terms are not cell averages

When the weights turn negative

With a full set of cell fixed effects the weights are non-negative. With linear covariates they need not be. Chattopadhyay and Zubizarreta (2023) show that an OLS coefficient on a binary treatment is a difference between a weighted mean of treated outcomes and a weighted mean of control outcomes, with the weight on person $i$ proportional to $D_i - \hat\ell_i$ (times $w_i$ in a survey-weighted regression), where $\hat\ell_i$ is the fitted value from a linear probability model of the treatment on the controls. A commuter whose fitted probability is below zero enters with the wrong sign, as if counted among the home workers.

In Chapter 4's headline specification this does not happen: fitted probabilities of working from home run from 0.9% to 22.1%, so every implied weight is positive. Add fixed effects for detailed occupation, industry, and state, as on the last rung of Chapter 4's ladder, and the additive linear probability model gives fitted values from −0.17 to 0.78. Then 8.6% of commuters, weighted by PERWT, carry negative weight (Figure 5.7). Their weights are small, together 0.7% of the total absolute weight, but they are concentrated where home work is rare: in Production, 21% of commuters are counted with the wrong sign.Strictly, even the cell model's linear age and sex terms push a few fitted probabilities just below zero (the lowest is −0.015, affecting 0.5% of commuters). The cell-level weights above are exact and remain non-negative.

Where the linear specification assigns negative weight

Share of each occupation group's commuters whose implied weight is negative, occupation + industry + state model

Show data

Category	Value
Production	21.0%
Transportation & moving	20.2%
Installation & repair	19.4%
Food preparation & serving	19.3%
Building & grounds	19.3%
Protective service	18.3%
Military specific	13.7%
Construction & extraction	11.7%
Healthcare support	10.8%
Farming, fishing & forestry	9.7%
Healthcare practitioners	8.4%
Education & library	7.3%
Personal care & service	6.5%
Office & admin support	4.5%
Sales	4.3%
Life, physical & social science	3.7%
Arts, design, media & sports	3.0%
Architecture & engineering	1.9%
Community & social service	1.9%
Management	1.0%

Figure 5.7.Under additive fixed effects some commuters get negative implied weight, most often in occupations where home work is rare.Note: Implied weight of observation i is PERWT_i(D_i − ℓ_i), where ℓ is the fitted value of a linear probability model for WFH with the regression's controls; a commuter with ℓ < 0 enters the comparison with the wrong sign. Groups under 0.5% not shown. In Chapter 4's headline model every fitted value lies between 0 and 1, so no weight is negative.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

Smaller groups get larger weights

Predict first

Only 13.7% of workers work from home. A population average gives the contrast for home workers that share of the weight and the contrast for commuters the rest. How much weight do you expect the regression to give each?

Słoczyński (2022) shows that an OLS coefficient on a binary treatment with additive controls can be written as follows; with weighted moments, the same holds for a survey-weighted regression:

\hat\beta = \omega_1\, \ATT + \omega_0\, \ATU, \qquad \omega_1 = \frac{(1-\rho)\, V_0}{\rho\, V_1 + (1-\rho)\, V_0}, \quad \omega_0 = 1 - \omega_1 ,

where $\omega_1$ is the implicit weight on the treated group's term (Słoczyński writes $w_1$ , the symbol the table and ledger keep), $\rho$ is the share treated, and $V_d$ is the variance of the linear propensity score (the fitted values of a linear probability model of the treatment on the controls) among people with treatment status $d$ . If the propensity score were equally variable in both groups, $\omega_1 = 1-\rho$ : OLS would give the treated group's effect the weight that belongs to the untreated group, and vice versa. The smaller group gets the larger weight.

Smaller groups get larger weights

Table of Sloczynski's decomposition for the unweighted and survey-weighted cell regressions: 14% of workers work from home, yet the contrast for home workers receives 83% of the regression's weight.
Regression	Coefficient	Share WFH (ρ)	Weight on ATT (ω₁)	ATT (APLE)	ATU (APLE)	APE
Unweighted (OLS)	0.122	14.0%	82.9%	0.118	0.137	0.135
Survey-weighted (WLS)	0.118	13.7%	83.2%	0.115	0.134	0.131

Figure 5.8.Home workers are a small minority, yet the home-worker term gets most of the regression's weight.Note: Cell-fixed-effect model. p(X) is the linear projection of WFH on the cells and the age–sex terms; ATT and ATU are Sloczynski's average partial linear effects, equal to the causal ATT and ATU only under unconfoundedness and linearity in p(X). Coefficient = ω₁·ATT + (1 − ω₁)·ATU exactly.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

In the survey-weighted cell regression (Figure 5.8), 13.7% of workers work from home and the weight on the home-worker term is 83%. The unweighted regression, in which 14.0% of records are home workers, gives 83%, and Chapter 4's model with education only gives 90%. That is the same pattern as Figure 5.3, where the regression sat next to the home-worker-share average.

Słoczyński's $\ATT$ and $\ATU$ are not the cell-based averages of Figure 5.3. They are average partial linear effects: within each group, log wage is projected on the propensity score by a straight line, and each term compares one group's mean with the other group's line evaluated at the first group's mean score. Here they are 0.115 for home workers and 0.134 for commuters, implying an average partial effect of 0.131, against cell-based averages of 0.120 and 0.083. The two pairs disagree even on which group's contrast is larger, because within groups mean log wage is not a straight-line function of the propensity score across occupation-by-education cells. The commuter term is the most exposed: it evaluates the home workers' line, fitted mostly where home work is common, at the low scores typical of commuters.

What the decomposition does establish is its weights. They depend only on the share working from home and on how much the propensity score varies within each group, and applied to the cell-based averages they give 0.114, close to the fitted coefficient. It does not establish a causal result. Reading the two terms as causal quantities would require unconfoundedness as well as linearity, and this descriptive analysis claims neither. Read Figure 5.8 for its weights, not for its ATT and ATU columns.

Checks on the decomposition

The decomposition is exact. The build checks the implementation against the numerical example in Słoczyński's paper and against simulations with known answers, and confirms that the fitted survey-weighted coefficient equals $\omega_1 \ATT + \omega_0 \ATU$ with the weighted-moment versions of $\rho$ , $V_1$ , and $V_0$ .

Analysis Ledger

Smaller groups get larger weights (Słoczyński decomposition)

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 cell-fixed-effect projection coefficient, re-expressed as ω₁·ATT + (1 − ω₁)·ATU, where ATT and ATU are average partial linear effects on WFH workers and commuters.
Estimator: Linear propensity score p(X) from WLS of WFH on cells and age–sex terms; within-group projections of log wage on p(X); weights from group shares and within-group variances of p(X).
Explicit weights: PERWT.
Implicit weights: ω₁ = (1 − ρ)V₀ / [ρV₁ + (1 − ρ)V₀] on the ATT; the smaller group gets the larger weight.
Source of randomness: The ACS sample and response process.
Variance estimator: SDR, 80 replicates, full recomputation; df = 79.
Assumptions: The decomposition is an in-sample identity; reading its components as ATT and ATU requires unconfoundedness and linearity in p(X), which this descriptive analysis does not claim.
Numbers reported: 83%
0.115log points
0.134log points
0.131log points

Note

A new case

A large cell holds many workers, but almost none of them works from home, and its contrast differs from the other cells'. How does the population-share average treat that cell, how does the regression treat it, and which summary is wrong?

One answer

The population-share average counts the cell at its full population share, so its contrast, which rests on the few home workers there and is imprecise, carries a lot of weight. The regression gives the cell almost nothing, because $p_c(1-p_c)$ is near zero. The two summaries therefore diverge, and neither is wrong: they answer different questions. The population-share average describes the whole workforce, including that cell's workers; the regression's summary mostly describes cells where home work is common. A report should say which question it answers and show the cell's contrast with its interval.

What we learned about the question

Among full-time wage and salary workers in 2024, the home–commute wage contrast is larger in occupations and education groups where working from home is common. The population-share average of the occupation-by-education cell contrasts is 0.088 log points; the survey-weighted regression with the same cells reports 0.118, close to the home-worker-share average. For the gender gap in the same cells the two summaries coincide, at −0.198 and −0.198, because the gap is about as large in the lopsided cells the regression discounts as in the mixed cells it favors.

What the method changed

Nothing about the sample or the survey weights changed. What changed is the weighting of cell contrasts: population shares versus the regression's $N_c p_c(1-p_c)$ , which gives 62% of the weight to ten cells holding 44% of workers and, in Słoczyński's decomposition, puts 83% of the weight on the home-worker term although home workers are 13.7% of workers.

What this analysis does not establish

A causal effect of working from home. Who works from home, and in which jobs, is selected in ways the cells and the age–sex profile do not remove; the home-worker-share and commuter-share averages borrow only the weighting schemes of the ATT and ATU.
An average that does not depend on the cells. The cells impose a common age–sex profile, 7 thin cells were merged, and a different partition, such as occupation by age, would give different contrasts and different weights.
Precise contrasts in small cells. Small cells have imprecise contrasts, and the population-share average counts them at full population weight.
A telework contrast. Chapter 4's construct caveat applies: TRANWORK records the usual means of getting to work, not any telework.
Słoczyński's ATT and ATU. Here they are linear-projection summaries that disagree with the cell-based averages, not estimates of the causal quantities.
A discrimination estimate. The gender contrast within cells is descriptive. Sorting into occupations, education, and home work is held fixed by the cells and controls, not explained, and the agreement of the two summaries says nothing about what the gap is a gap of.