---
title: "fetwfe: A Package for Fused Extended Two-Way Fixed Effects"
author: "Gregory Faletto"
date: "`r Sys.Date()`"
output:
  rmarkdown::html_vignette:
    toc: true
    math_method: "mathml"
    output_file: "FETWFE_vignette.html"
vignette: >
  %\VignetteIndexEntry{fetwfe: A Package for Fused Extended Two-Way Fixed Effects}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
  %\VignetteDepends{dplyr, bacondecomp}
---

```{r setup, include=FALSE}
knitr::opts_chunk$set(echo = TRUE)
library(dplyr)
```

# Introduction

If you understand the basic idea of what difference-in-differences with staggered adoptions is, all you need to know about fused extended two-way fixed effects (FETWFE) to get started using the `{fetwfe}` package is this: given an appropriately formatted panel data set, `fetwfe()` will give you an estimate of the overall average treatment effect on the treated units, the average treatment effect within each cohort, and standard errors for each of these estimates.

Feel free to skip to the "Package Usage" section if you want to jump right in to using the package. In the next "Background" subsection, you can read a little more background information on the methodology if you'd like.

## Background

This vignette is written under the assumption that you're at least vaguely familiar with developments in [difference-in-differences](https://en.wikipedia.org/wiki/Difference_in_differences) with [staggered adoptions](https://mike-data-analysis.share.connect.posit.cloud/sec-difference-in-differences.html#sec-multiple-periods-and-variation-in-treatment-timing) 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](https://mike-data-analysis.share.connect.posit.cloud/sec-difference-in-differences.html#sec-two-way-fixed-effects) 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](https://arxiv.org/abs/2312.05985)**

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](https://gregoryfaletto.com/2023/12/13/new-paper-fused-extended-two-way-fixed-effects-for-difference-in-differences-with-staggered-adoptions/) announcing the paper.
- [Some slides](https://gregoryfaletto.com/2024/02/11/presentation-on-fused-extended-two-way-fixed-effects/) I made for a presentation on FETWFE.
- Another [blog post](https://gregoryfaletto.com/2025/01/03/new-r-fetwfe-package-implementing-fused-extended-two-way-fixed-effects/) 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.**

# Package Usage

The package provides a single exported function, `fetwfe()`, which implements the FETWFE estimator. Its primary arguments include:

- **`pdata`**: A data frame in panel (long) format.
- **`time_var`**: A character string specifying the name of the time period variable.
- **`unit_var`**: A character string specifying the unit (e.g. state, firm) variable.
- **`treatment`**: A character string specifying the treatment indicator variable (which must be an absorbing binary indicator).
- **`response`**: A character string specifying the response (outcome) variable.
- **`covs`**: A character vector of covariate names (typically time-invariant or the pre-treatment values), if applicable.
- **Additional tuning parameters:** such as the tuning parameter for the bridge penalty (controlled via argument `q`), the geometry of the fusion penalty (`fusion_structure`, either the default `"cohort"` or `"event_study"`, or a fully custom `fusion_matrix`), and options for verbosity, standard error calculation, and so on.

The function returns a list containing, for example, the estimated overall average treatment effect, cohort-specific treatment effects, standard errors (when available), and various diagnostic quantities.

You can get the full documentation details by using `?fetwfe` in R when you have the package loaded.

**The workflow in three moves.** Working with a fit comes down to:

1. **Fit** --- call `fetwfe()` on your panel.
2. **Extract** --- pull out the treatment effects by the view you want: `cohortStudy()` (per adoption cohort), `eventStudy()` (per time since treatment), or `cohortTimeATTs()` (the fully disaggregated cohort-by-time cells); the overall ATT is on the fit as `att_hat` / `att_se`.
3. **Plot** --- visualize them with `plot(fit, type = "event_study")` or `plot(fit, type = "catt")`.

The rest of this vignette expands on each of these.

For a detailed discussion of how the package's standard errors are computed, the assumptions they rely on, and an experimental cluster-robust option (`se_type = "cluster"`) suitable as a sensitivity check, see the companion vignette `inference_vignette`. Two further companion vignettes cover family-wise simultaneous confidence bands (`simultaneousCIs()`) and the selection-consistency zero-effect test (`simultaneous_cis_vignette`), and high-dimensional debiased inference via `debiasedATT()` (`high_dimensional_vignette`).

In the next sections, we'll walk through examples of how `fetwfe()` is used.

## 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](https://asjadnaqvi.github.io/)'s helpful [website for DiD estimators](https://asjadnaqvi.github.io/DiD/docs/code_r). Thanks for sharing the code publicly!**

```{r}
# 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)
```

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()`.

```{r}
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)
```

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

```{r}

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)
```

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.

By default the fusion penalty uses the within-/between-cohort structure, but you can instead fuse treatment effects by event time (time since treatment) with `fusion_structure = "event_study"`, or supply a fully custom fusion matrix. For when to prefer each, see the vignette ["Choosing a fusion structure: cohort vs. event-study penalties"](fusion_structure_vignette.html) (`vignette("fusion_structure_vignette", package = "fetwfe")`).

## A "Real Data" Example

Next I illustrate FETWFE in an empirical context. I'll use the `divorce` data set from the `bacondecomp` package, which comes from the study by Stevenson and Wolfers (2006) of unilateral ("no-fault") divorce laws and female suicide rates. It is a balanced state-year panel covering 51 states over 1964-1996, with the reforms adopted in a staggered, scattered fashion across states. This is the empirical application in Section 8.2 of Faletto (2025).

```{r}
library(bacondecomp)  # for the example data

# Load the data and restrict to the female subset (`sex == 2`), the sample
# analyzed by Stevenson and Wolfers (2006).
data(divorce)
divorce_f <- divorce[divorce$sex == 2, ]

# - 'year' is the time period, 'st' is the unit (state) identifier;
# - 'changed' is already an absorbing 0/1 divorce-reform indicator;
# - 'suiciderate_elast_jag' is the elasticity-scaled female suicide rate.
# Three covariates enter as controls. fetwfe() automatically drops the 9 states
# already treated by 1964 (first-period-treated) and `murderrate` (missing in
# 1964 for one state); both are reported as warnings, suppressed here for
# brevity. The noise variances are supplied (precomputed by REML) to keep the
# call fast.
res <- suppressWarnings(fetwfe(
    pdata = divorce_f,
    time_var = "year",
    unit_var = "st",
    treatment = "changed",
    covs = c("murderrate", "lnpersinc", "afdcrolls"),
    response = "suiciderate_elast_jag",
    sig_eps_sq = 0.0344,
    sig_eps_c_sq = 0.1507,
    add_ridge = TRUE,
    q = 0.5
    ))

summary(res)

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

# 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)

# Cohort average treatment effects and confidence intervals (in percentage
# point units). Several cohorts are pruned to exactly zero by the selection.

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
```

FETWFE estimates that adopting a unilateral divorce law is associated with roughly a -6% change in the elasticity-scaled female suicide rate, with a 95% confidence interval that excludes zero. The sign and order of magnitude match Stevenson and Wolfers (2006), who found that these reforms reduced female suicide. Unlike a fully fused fit, FETWFE's selection step retains *heterogeneous* cohort effects here: several adoption cohorts are pruned to exactly zero while the rest carry nonzero, varying estimates, so the rows of `res$catt_df` are no longer all identical.

Unlike the simulated example above, this fit includes covariates (the `covs` argument). Because FETWFE regularizes the cohort-interacted design through its bridge penalty, it accommodates covariates even on this panel's scattered, single-state adoption cohorts -- a setting in which an unregularized two-way fixed effects fit would be rank-deficient.

## 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`.

```{r zero-effect-demo, message = FALSE}
set.seed(2026)
sim <- genCoefs(G = 3, T = 6, d = 2, density = 0.5, eff_size = 2)
# sim was built without a seed, so draw from the ambient generator (seeded
# just above); seed = NA selects the ambient generator silently.
dat <- simulateData(sim, N = 120, sig_eps_sq = 1, sig_eps_c_sq = 0.5, seed = NA)
res_demo <- fetwfeWithSimulatedData(dat, verbose = FALSE)
res_demo$catt_df
```

`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, per-event-time, and per-(cohort, 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()`:

```{r cohort-study-demo, message = FALSE}
cs <- cohortStudy(res_demo)
cs
```

```{r cohort-study-tidy, eval = requireNamespace("broom", quietly = TRUE)}
broom::tidy(cs)
```

`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.

For the *fully disaggregated* effects --- one row per `(cohort, time)` cell, with no averaging over either axis --- use `cohortTimeATTs()`. This is the finest-grained view, sitting below `cohortStudy()` (which averages each cohort's cells over time) and `eventStudy()` (which averages over cohorts at each event time):

```{r cohort-time-atts-demo, message = FALSE}
cta <- cohortTimeATTs(res_demo)
head(cta)
```

Its standard errors are the per-cell regression standard errors, recomputed from the fit exactly as `eventStudy()` does (each cell is a single coefficient, so the cohort-probability sampling variance that contributes to the aggregated SEs is identically zero here). Confidence intervals and p-values are pointwise; a cell the fusion penalty zeroed out has `estimate = 0`, `se = 0`, and `selected = FALSE`. `broom::tidy()` works here too (`term = "cohort_<g>_time_<t>"`). For simultaneous (family-wise) bands over the `(cohort, time)` cell family, use `simultaneousCIs(res_demo, family = "all_post_treatment")`.

```{r cohort-time-atts-tidy, eval = requireNamespace("broom", quietly = TRUE)}
head(broom::tidy(cta))
```

## Plotting the effects

The built-in `plot()` method draws the effects directly --- the event-study view (`type = "event_study"`, the default) or the per-cohort view (`type = "catt"`), each with confidence intervals:

```{r plot-event-study, fig.width = 7, fig.height = 4, eval = requireNamespace("ggplot2", quietly = TRUE)}
plot(res_demo, type = "event_study")
```

Reading it: each point is the average treatment effect at a given number of periods since adoption (the *event time*), with its confidence interval. The plot begins at event time 0 (treatment onset) and traces how the effect evolves over the post-treatment periods --- FETWFE does not estimate pre-treatment placebo coefficients (they are out of scope; see `?eventStudy`), so there is no pre-trends panel here. Swap in `type = "catt"` for the per-adoption-cohort effects, where any cohort the fusion penalty pooled to zero appears pinned exactly at zero --- a visual signature of the fusion. (The `broom` route in the next section is the do-it-yourself alternative when you want full `ggplot2` control.)

## Tidy outputs with broom

If you use the tidyverse, the fitted object also plays nicely with the [broom](https://broom.tidymodels.org/) 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):

```{r tidy-demo, eval = requireNamespace("broom", quietly = TRUE)}
library(broom)

tidy_res <- tidy(result)
tidy_res
```

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

```{r glance-demo, eval = requireNamespace("broom", quietly = TRUE)}
glance(result)
```

`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):

```{r augment-demo, eval = requireNamespace("broom", quietly = TRUE)}
head(augment(result, data = pdata))
```

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

```{r tidy-plot, eval = requireNamespace("broom", quietly = TRUE) && requireNamespace("ggplot2", quietly = TRUE) && requireNamespace("dplyr", quietly = TRUE)}
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.

# Conclusion

This should be enough to get you started using `fetwfe()` on your own data. Please feel free to [reach out](https://gregoryfaletto.com/about/) if you have any questions or feedback or run into any issues using the package. You can also [create an issue](https://github.com/gregfaletto/fetwfePackage/issues) if you think there’s a bug in the package or you’d like to request a feature. **Thanks so much for checking out the package!**

# References

- Cheng, C., & Hoekstra, M. (2013). Does Strengthening Self-Defense Law Deter Crime or Escalate Violence? Evidence from Expansions to Castle Doctrine. *Journal of Human Resources*, 48(3), 821-854.
- Faletto, G. (2025). Fused Extended Two-Way Fixed Effects for Difference-in-Differences with Staggered Adoptions. [arXiv preprint arXiv:2312.05985](https://arxiv.org/abs/2312.05985).
- Flack, E., & Jee, E. (2020). bacondecomp: Goodman-Bacon Decomposition. R package version 0.1.1. [https://CRAN.R-project.org/package=bacondecomp](https://CRAN.R-project.org/package=bacondecomp).
- Kock, A. B. (2013). Oracle inequalities for high-dimensional nonparametric models. *Econometric Theory*.

