Background

This vignette is written under the assumption that you’re at least vaguely familiar with developments in difference-in-differences with staggered adoptions since about 2018. Just to make sure we’re on the same page, the brief recap is:

• Historically, under staggered adoptions researchers used the standard two-way fixed effects estimator and interpreted the coefficient on the treatment dummy as an average treatment effect on the treated units.
• In the late 2010s, econometricians formally checked what this estimator was doing and found that in fact, this coefficient was not any kind of reasonable average treatment effect estimator.
• Since then, a number of new difference-in-differences estimators that are asymptotically unbiased under staggered adoptions have been developed.

The estimator in this package, fused extended two-way fixed effects (FETWFE), is one of those asymptotically unbiased estimators. Of course, I made this estimator because I think FETWFE brings something to the table that the others don’t. Here’s a brief summary on that:

One issue with these estimators has been that they’ve worked so hard to be unbiased that they are inefficient (in the language of econometrics), or high-variance (in the language of machine learning). These estimators add extra parameters in order to remove bias, but estimating extra parameters means you have less data per parameter and your estimates are noisier.

In machine learning, creating a more flexible estimator with lots of parameters and then finding that it is too high variance (that is, it overfits) is a familiar issue. The most common solution has been regularization.

You could just add \(\ell_2\) or \(\ell_1\) regularization to a difference-in-differences regression estimator and probably see an improvement in your efficiency, but FETWFE does something more sophisticated than that. (Plus, that approach wouldn’t allow you to get valid standard errors for your treatment effect estimates, but FETWFE does.) Qualitatively, FETWFE uses machine learning to learn which of these added parameters were actually unnecessary to add, and then takes them back out in order to improve efficiency.

That’s all the description I’ll give you in this vignette. You can learn all of the details in the paper on arXiv:

Fused Extended Two-Way Fixed Effects for Difference-in-Differences With Staggered Adoptions

If you want to learn a little more before you dive into the full paper, here are some other resources with descriptions of the methodology that provide a little more detail than this vignette:

• My blog post announcing the paper.
• Some slides I made for a presentation on FETWFE.
• Another blog post focused on what this package is doing under the hood.

But the headline summary of what fused extended two-way fixed effects brings to the table in a crowded field of estimators is: fused extended two-way fixed effects is not only unbiased, it also uses machine learning to maximize efficiency (minimize variance). Further, unlike many machine learning estimators, fused extended two-way fixed effects gives you valid standard errors for the treatment effect estimates.

Simulated Data Example

I’ll start illustrating how to use fetwfe() by using a simulated data set. The example below simulates a balanced panel with 20 time periods and 30 individuals. Each individual is assigned at random to one of five cohort levels; with the randomly drawn treatment timing, three of those levels resolve to cohorts that are actually treated within the panel window and the rest are never treated.

In the simulation, each individual is assigned a random cohort (which determines the timing of treatment) and three time-invariant covariates are generated. The response variable is constructed so that, after treatment, its evolution depends on a treatment effect (which varies by cohort) and a linear trend, plus the covariates and some random noise.

Below is the complete code for simulating the data, converting it into the required pdata format, and running the fetwfe() function.

I borrowed some of the below code from Asjad Naqvi’s helpful website for DiD estimators. Thanks for sharing the code publicly!

# Set seed for reproducibility
set.seed(123456L)

# 20 time periods, 30 individuals, and 5 cohort levels
tmax = 20; imax = 30; nlvls = 5

dat =
  expand.grid(time = 1:tmax, id = 1:imax) |>
  within({
    cohort      = NA
    effect      = NA
    first_treat = NA
    cov1        = NA
    cov2        = NA
    cov3        = NA
    for (chrt in 1:imax) {
      cohort = ifelse(id==chrt, sample.int(nlvls, 1), cohort)
    }
    for (lvls in 1:nlvls) {
      effect      = ifelse(cohort==lvls, sample(2:10, 1), effect)
      first_treat = ifelse(cohort==lvls, sample(1:(tmax+6), 1), first_treat)
    }
    # three time-invariant covariates: one value per individual,
    # drawn AFTER the timing loop so first_treat is unchanged
    for (chrt in 1:imax) {
      cov1 = ifelse(id==chrt, rnorm(1), cov1)
      cov2 = ifelse(id==chrt, rnorm(1), cov2)
      cov3 = ifelse(id==chrt, rnorm(1), cov3)
    }
    first_treat = ifelse(first_treat>tmax, Inf, first_treat)
    treat       = time >= first_treat
    rel_time    = time - first_treat
    y           = id + time + ifelse(treat, effect*rel_time, 0) +
                  0.5*cov1 - 0.7*cov2 + 1.2*cov3 + rnorm(imax*tmax)
    rm(chrt, lvls, cohort, effect)
  })

head(dat)

##   time id         y rel_time treat       cov3      cov2       cov1 first_treat
## 1    1  1 1.3403587     -Inf FALSE -0.5019485 0.1582893 -0.8962503         Inf
## 2    2  1 0.2842639     -Inf FALSE -0.5019485 0.1582893 -0.8962503         Inf
## 3    3  1 3.7222320     -Inf FALSE -0.5019485 0.1582893 -0.8962503         Inf
## 4    4  1 3.6309664     -Inf FALSE -0.5019485 0.1582893 -0.8962503         Inf
## 5    5  1 6.6701082     -Inf FALSE -0.5019485 0.1582893 -0.8962503         Inf
## 6    6  1 6.0159986     -Inf FALSE -0.5019485 0.1582893 -0.8962503         Inf

The simulated data (dat) now has columns for time, id, a treatment indicator (treat), three time-invariant covariates (cov1, cov2, cov3), and a response variable (y). Next, we convert this data into the panel data format required by fetwfe().

library(dplyr)

# Specify column names for the pdata format
time_var <- "time"       # Column for the time period
unit_var <- "unit"       # Column for the unit identifier
treatment <- "treated"   # Column for the treatment dummy indicator
response <- "response"   # Column for the response variable

# Convert the dataset
pdata <- dat |>
  mutate(
    # Rename id to unit and convert to character
    {{ unit_var }} := as.character(id),
    # Ensure treatment dummy is 0/1
    {{ treatment }} := as.integer(treat),
    # Rename y to response
    {{ response }} := y
  ) |>
  select(
    {{ time_var }}, {{ unit_var }}, {{ treatment }}, {{ response }},
    cov1, cov2, cov3
  )

# Preview the resulting pdata dataframe
head(pdata)

##   time unit treated  response       cov1      cov2       cov3
## 1    1    1       0 1.3403587 -0.8962503 0.1582893 -0.5019485
## 2    2    1       0 0.2842639 -0.8962503 0.1582893 -0.5019485
## 3    3    1       0 3.7222320 -0.8962503 0.1582893 -0.5019485
## 4    4    1       0 3.6309664 -0.8962503 0.1582893 -0.5019485
## 5    5    1       0 6.6701082 -0.8962503 0.1582893 -0.5019485
## 6    6    1       0 6.0159986 -0.8962503 0.1582893 -0.5019485

Now that pdata is properly formatted, we run the FETWFE estimator on the simulated data.

library(fetwfe)

# Run the FETWFE estimator on the simulated data
result <- fetwfe(
  pdata = pdata,                        # The panel dataset
  time_var = "time",                    # The time variable
  unit_var = "unit",                    # The unit identifier
  treatment = "treated",                # The treatment dummy indicator
  response = "response",                # The response variable
  covs = c("cov1", "cov2", "cov3")      # The time-invariant covariates
)

# Display the average treatment effect estimates
summary(result)

## Summary of Fused Extended Two-Way Fixed Effects
## ================================================
## 
## Overall ATT: 30.4626  (SE = 4.1889, p = 3.535e-13, 95% CI = [22.2525, 38.6726])
## Selected: TRUE
## 
## CATT (preview) [simultaneous 95% CI]:
##  cohort estimate        se   ci_low  ci_high p_value selected
##       7 58.84753 0.2165463 58.33381 59.36124       0     TRUE
##      11 14.00563 0.2344958 13.44934 14.56193       0     TRUE
##      12 20.01437 0.1830776 19.58006 20.44869       0     TRUE
## 
## Event Study (preview) [simultaneous 95% CI]:
##  event_time n_cohorts   estimate        se      ci_low    ci_high      p_value
##           0         3   1.283027 0.2459480   0.6170528   1.949000 1.303408e-06
##           1         3   6.396242 0.5106127   5.0136135   7.778871 0.000000e+00
##           2         3  11.291839 0.9463132   8.7294283  13.854250 0.000000e+00
##           3         3  18.229410 1.4991523  14.1700307  22.288790 0.000000e+00
##           4         3  23.403446 2.0417170  17.8749187  28.931973 0.000000e+00
##           5         3  29.525165 2.7279717  22.1384088  36.911921 0.000000e+00
##           6         3  33.886714 2.9253379  25.9655329  41.807895 0.000000e+00
##           7         3  39.974990 3.6601035  30.0642229  49.885757 0.000000e+00
##           8         3  46.988747 4.0877163  35.9200972  58.057397 0.000000e+00
##           9         2  60.016166 8.2868965  37.5770456  82.455286 2.591150e-12
##          10         1  89.791179 0.5431113  88.3205512  91.261806 0.000000e+00
##          11         1  98.188340 0.5431113  96.7177126  99.658968 0.000000e+00
##          12         1 107.809727 0.4963810 106.4656349 109.153819 0.000000e+00
##          13         1 118.370341 0.4963810 117.0262492 119.714434 0.000000e+00
##          14         0   0.000000        NA          NA         NA           NA
##          15         0   0.000000        NA          NA         NA           NA
##          16         0   0.000000        NA          NA         NA           NA
##          17         0   0.000000        NA          NA         NA           NA
##          18         0   0.000000        NA          NA         NA           NA
## 
## Model Details:
##   Units (N)           : 30
##   Time periods (T)    : 20
##   Treated cohorts (G) : 3
##   Covariates (d)      : 3
##   Features (p)        : 223
##   Selected size       : 43
##   Lambda*             : 0.0309

When you run this code, the function internally performs all the necessary data preparation, applies the fusion penalty via a bridge regression (using the grpreg package), and returns a list with overall and cohort-specific treatment effect estimates, standard errors (if available), and additional diagnostics.

A “Real Data” Example

Next I illustrate FETWFE in an empirical context. I’ll use the castle data set from the bacondecomp package, which comes from the study by Cheng and Hoekstra (2013) of castle-doctrine (“stand-your-ground”) laws and homicide. It is a balanced state-year panel covering all 50 states over 2000-2010, with the laws adopted in a staggered fashion across states from 2005 to 2009.

library(bacondecomp)  # for the example data

# Load the example data
data(castle)

# Response: the log homicide rate, so the ATT reads roughly as a
# percentage change in the homicide rate.
castle$l_homicide <- log(castle$homicide)

# Treatment indicator. `cdl` records the share of the year the
# castle-doctrine law was in effect; a state is treated from the year its
# law took effect, so we binarize with `cdl > 0`. `fetwfe()` requires the
# treatment column to be an integer 0/1 absorbing indicator.
castle$treated <- as.integer(castle$cdl > 0)

# Call the estimator. Here
# - 'year' is the time period variable (an integer),
# - 'state' is the unit identifier,
# - 'treated' is the absorbing treatment indicator.
res <- fetwfe(
    pdata = castle,
    time_var = "year",
    unit_var = "state",
    treatment = "treated",
    response = "l_homicide"
    )

summary(res)

## Summary of Fused Extended Two-Way Fixed Effects
## ================================================
## 
## Overall ATT: 0.0522  (SE = 0.0277, p = 0.05969, 95% CI = [-0.0021, 0.1065])
## Selected: TRUE
## 
## CATT (preview) [simultaneous 95% CI]:
##  cohort   estimate         se       ci_low   ci_high    p_value selected
##    2005 0.05218228 0.02771106 -0.002141695 0.1065063 0.05968889     TRUE
##    2006 0.05218228 0.02771106 -0.002141695 0.1065063 0.05968889     TRUE
##    2007 0.05218228 0.02771106 -0.002141695 0.1065063 0.05968889     TRUE
##    2008 0.05218228 0.02771106 -0.002141695 0.1065063 0.05968889     TRUE
##    2009 0.05218228 0.02771106 -0.002141695 0.1065063 0.05968889     TRUE
## 
## Event Study (preview) [simultaneous 95% CI]:
##  event_time n_cohorts   estimate         se       ci_low   ci_high    p_value
##           0         5 0.05218228 0.02771106 -0.002142757 0.1065073 0.05968889
##           1         5 0.05218228 0.02771106 -0.002142757 0.1065073 0.05968889
##           2         4 0.05218228 0.02771106 -0.002142757 0.1065073 0.05968889
##           3         3 0.05218228 0.02771106 -0.002142757 0.1065073 0.05968889
##           4         2 0.05218228 0.02771106 -0.002142757 0.1065073 0.05968889
##           5         1 0.05218228 0.02771106 -0.002142757 0.1065073 0.05968889
##           6         0 0.00000000         NA           NA        NA         NA
##           7         0 0.00000000         NA           NA        NA         NA
##           8         0 0.00000000         NA           NA        NA         NA
##           9         0 0.00000000         NA           NA        NA         NA
## 
## Model Details:
##   Units (N)           : 50
##   Time periods (T)    : 11
##   Treated cohorts (G) : 5
##   Covariates (d)      : 0
##   Features (p)        : 35
##   Selected size       : 5
##   Lambda*             : 0.0013

# Average treatment effect on the treated units (in percentage point
# units)
100 * res$att_hat

## [1] 5.218228

# Conservative 95% confidence interval for ATT (in percentage point units)

low_att <- 100 * (res$att_hat - qnorm(1 - 0.05 / 2) * res$att_se)
high_att <- 100 * (res$att_hat + qnorm(1 - 0.05 / 2) * res$att_se)

c(low_att, high_att)

## [1] -0.2130406 10.6494970

# Cohort average treatment effects and confidence intervals (in percentage
# point units)

catt_df_pct <- res$catt_df
catt_df_pct[["estimate"]] <- 100 * catt_df_pct[["estimate"]]
catt_df_pct[["se"]] <- 100 * catt_df_pct[["se"]]
catt_df_pct[["ci_low"]] <- 100 * catt_df_pct[["ci_low"]]
catt_df_pct[["ci_high"]] <- 100 * catt_df_pct[["ci_high"]]

catt_df_pct

##   cohort estimate       se     ci_low  ci_high    p_value selected
## 1   2005 5.218228 2.771106 -0.2141695 10.65063 0.05968889     TRUE
## 2   2006 5.218228 2.771106 -0.2141695 10.65063 0.05968889     TRUE
## 3   2007 5.218228 2.771106 -0.2141695 10.65063 0.05968889     TRUE
## 4   2008 5.218228 2.771106 -0.2141695 10.65063 0.05968889     TRUE
## 5   2009 5.218228 2.771106 -0.2141695 10.65063 0.05968889     TRUE

FETWFE estimates that adopting a castle-doctrine law is associated with roughly a +5.5% change in the homicide rate. The sign and rough magnitude are consistent with Cheng and Hoekstra (2013), who found that these laws increased homicide. FETWFE fused all five adoption cohorts (those adopting in 2005 through 2009) to a single common effect, so each row of res$catt_df reports the same estimate.

This example is covariate-free; castle’s smallest cohorts contain a single state, so the design cannot support additional covariates. Covariate handling is illustrated in the simulated example above — the covs argument is optional.

Testing the zero-effect null

Empirical users of difference-in-differences routinely want to ask “is this treatment effect statistically distinguishable from zero?”. The confidence intervals in catt_df answer that implicitly — a CI that excludes 0 corresponds to rejecting H_0: tau = 0 at level alpha (family-wise across cohorts under the default simultaneous bands). The package also surfaces a per-cohort p_value that matches whichever interval is displayed: under the default ci_type = "simultaneous" it is the single-step max-T multiplicity-adjusted p-value (the exact dual of the simultaneous band, so the CI-excludes-0 test and p_value < alpha always agree; version 1.18.0), and under ci_type = "pointwise" the two-sided Wald 2 * pnorm(-|estimate / se|). The overall-ATT att_p_value is the scalar Wald p-value (a single effect, no multiplicity). For fetwfe() and betwfe() there is also a selected logical flag, both at the overall ATT level and per-cohort.

For fetwfe(), the selected flag carries something stronger than the usual CI test. Under the restriction selection consistency theorem (Theorem 6.2 in the paper), when the true tau_ATT(g, t) = 0, the estimator returns exactly 0 with probability tending to 1. The package’s interpretation: when selected = FALSE for a cohort, the asymptotic conclusion is that the truth is zero — a strictly stronger statement than what a standard CI provides. The package surfaces this as an asymptotic 100% confidence interval of {0} for selected-out cohorts. For selected-out cohorts, p_value is reported as NA — the inferential content lives in selected.

set.seed(2026)
sim <- genCoefs(G = 3, T = 6, d = 2, density = 0.5, eff_size = 2)
dat <- simulateData(sim, N = 120, sig_eps_sq = 1, sig_eps_c_sq = 0.5)
res_demo <- fetwfeWithSimulatedData(dat, verbose = FALSE)
res_demo$catt_df

##   cohort   estimate         se    ci_low    ci_high      p_value selected
## 1      2 -0.4137526 0.07754859 -0.586916 -0.2405893 1.906689e-07     TRUE
## 2      3 -1.6817852 0.12781952 -1.967202 -1.3963687 0.000000e+00     TRUE
## 3      4  0.0000000 0.00000000  0.000000  0.0000000           NA    FALSE

betwfe() uses bridge regression directly on the coefficients (rather than on the fused restrictions used by fetwfe()). Under the bridge oracle property of Kock (2013), selected = FALSE for betwfe() is an analogous asymptotic statement that the truth is zero, under a sparsity assumption different from the one Theorem 6.2 establishes for fetwfe(). For etwfe() and twfeCovs(), neither has a selection step, so the selected column is omitted; the p_value column is a standard post-OLS t-test.

It is worth keeping in mind that the asymptotic 100% CI of {0} interpretation is an N -> infinity statement. In small samples, it may be wise to read selected and p_value together rather than relying on either alone. In particular, even when selected = TRUE, the standard CI may cover zero — the selection step says “the truth is nonzero” but the magnitude CI says “could be zero”.

Per-cohort and per-event-time accessors

The per-cohort estimates in res$catt_df are also reachable through cohortStudy(), a function-style accessor parallel to eventStudy() for event-time effects. The two are documented and discoverable in the same way (?cohortStudy, ?eventStudy), and both return a data frame that plays nicely with broom::tidy():

cs <- cohortStudy(res_demo)
cs

##   cohort   estimate         se    ci_low    ci_high      p_value selected
## 1      2 -0.4137526 0.07754859 -0.586916 -0.2405893 1.906689e-07     TRUE
## 2      3 -1.6817852 0.12781952 -1.967202 -1.3963687 0.000000e+00     TRUE
## 3      4  0.0000000 0.00000000  0.000000  0.0000000           NA    FALSE

broom::tidy(cs)

##       term   estimate  std.error  statistic      p.value  conf.low  conf.high
## 1 cohort_2 -0.4137526 0.07754859  -5.335399 1.906689e-07 -0.586916 -0.2405893
## 2 cohort_3 -1.6817852 0.12781952 -13.157499 0.000000e+00 -1.967202 -1.3963687
## 3 cohort_4  0.0000000 0.00000000         NA           NA  0.000000  0.0000000
##   selected
## 1     TRUE
## 2     TRUE
## 3    FALSE

cohortStudy() is a pass-through on res$catt_df — same rows, same columns — with one extra S3 class on top ("cohortStudy") that dispatches tidy() to a broom-shape translator. The catt_df class underneath is preserved, so the helpful-error layer on the old Title-Case column names still fires.

Tidy outputs with broom

If you use the tidyverse, the fitted object also plays nicely with the broom package. tidy() reshapes the overall ATT and per-cohort CATTs into a long data frame with the standard term / estimate / std.error / statistic / p.value / conf.low / conf.high columns, plus a selected flag (see “Testing the zero-effect null” below):

library(broom)

tidy_res <- tidy(result)
tidy_res

##        term estimate std.error  statistic      p.value conf.low conf.high
## 1       ATT 30.46257 4.1888700   7.272264 3.535113e-13 22.25254  38.67260
## 2  Cohort 7 58.84753 0.2165463 271.754951 0.000000e+00 58.33381  59.36124
## 3 Cohort 11 14.00563 0.2344958  59.726581 0.000000e+00 13.44934  14.56193
## 4 Cohort 12 20.01437 0.1830776 109.321780 0.000000e+00 19.58006  20.44869
##   selected
## 1     TRUE
## 2     TRUE
## 3     TRUE
## 4     TRUE

glance() returns a one-row model-level summary — panel-shape counts, the bridge-regression tuning, the variance components, and the inference settings:

glance(result)

##   nobs n_units n_periods n_cohorts n_covs n_features lambda_star
## 1  600      30        20         3      3        223  0.03085978
##   lambda_star_model_size lambda_selection cv_folds cv_seed sig_eps_sq
## 1                     43               cv       10     600  0.9715147
##   sig_eps_c_sq alpha se_type indep_counts_used
## 1     90.72021  0.05 default             FALSE

augment() appends .fitted and .resid columns to your panel (auto-aligned to the design the estimator actually fit on, so the same raw pdata you handed to fetwfe() works):

head(augment(result, data = pdata))

##   time unit treated  response       cov1      cov2       cov3  .fitted
## 1    1    1       0 1.3403587 -0.8962503 0.1582893 -0.5019485 37.87288
## 2    2    1       0 0.2842639 -0.8962503 0.1582893 -0.5019485 39.05406
## 3    3    1       0 3.7222320 -0.8962503 0.1582893 -0.5019485 39.71505
## 4    4    1       0 3.6309664 -0.8962503 0.1582893 -0.5019485 40.88863
## 5    5    1       0 6.6701082 -0.8962503 0.1582893 -0.5019485 41.89930
## 6    6    1       0 6.0159986 -0.8962503 0.1582893 -0.5019485 43.22619
##      .resid
## 1 -36.53252
## 2 -38.76980
## 3 -35.99282
## 4 -37.25766
## 5 -35.22919
## 6 -37.21019

From there, the tidied output goes straight into ggplot2 or modelsummary without needing to read the package’s class documentation:

library(ggplot2)
tidy_res |> 
  dplyr::filter(term != "ATT") |>
  # Remove non-digits to extract the number, then reorder the factor levels
  dplyr::mutate(term = reorder(term, as.numeric(gsub("\\D", "", term)))) |>
  ggplot(aes(x = term, y = estimate)) +
  geom_pointrange(aes(ymin = conf.low, ymax = conf.high)) +
  geom_hline(yintercept = 0, linetype = "dashed") +
  labs(x = NULL, y = "Cohort treatment effect", title = "Cohort ATTs")

See the simulation vignette for an example of how you can use functions in the FETWFE package to simulate panel data.