Module 1: TWFE Diagnosed

---

# Course Map

<table>
<tr><th>#</th><th>Module</th><th>Status</th></tr>
<tr><td><b>1</b></td><td><b>TWFE Diagnosed: Goodman-Bacon and the Zoo of 2x2s</b> <i>(you are here)</i></td><td>current</td></tr>
<tr><td>2</td><td><a href="../module-02/slides.html">Heterogeneity-Robust DiD: CS, SA, BJS, dCDH</a></td><td>done</td></tr>
<tr><td>3</td><td><a href="../module-03/slides.html">Honest DiD: Sensitivity Bounds for Parallel Trends</a></td><td>done</td></tr>
<tr><td>4</td><td><a href="../module-04/slides.html">Synthetic Control: Estimator, Inference, Variants</a></td><td>done</td></tr>
<tr><td>5</td><td><a href="../module-05/slides.html">Synthetic DiD and the Bridge from SC to DiD</a></td><td>done</td></tr>
<tr><td>6</td><td><a href="../module-06/slides.html">Causal Forest: Honest Splitting and Asymptotics</a></td><td>done</td></tr>
<tr><td>7</td><td><a href="../module-07/slides.html">Policy Learning: From HTE to Deployment Rules</a></td><td>done</td></tr>
<tr><td>8</td><td><a href="../module-08/slides.html">Matrix Completion and the Modern Panel Toolbox</a></td><td>done</td></tr>
</table>

---

# This Module

The experimentation-refresher tour showed *that* TWFE breaks under staggered
adoption. This module shows *exactly how*, with the full Goodman-Bacon (2021)
decomposition.

**You will be able to:**

1. State the TWFE estimand under staggered timing as a weighted average of
   all possible 2x2 DiD comparisons.
2. Derive and compute the Bacon weights by hand, and replicate
   `bacondecomp::bacon()` to machine precision.
3. Explain the forbidden comparison and show when it flips signs.
4. Say precisely when TWFE is still fine, and defend that in an interview.

**Running application:** staggered rollout of a driver zone-notification
feature across 30 cities, 60 weeks, three adoption cohorts plus 6
never-treated cities.

---

# The Estimator Under the Microscope

The two-way fixed effects (TWFE) regression:

`$$y_{it} = \alpha_i + \lambda_t + \beta^{DD} D_{it} + \varepsilon_{it}$$`

- `$\alpha_i$`: unit fixed effects. `$\lambda_t$`: period fixed effects.
- `$D_{it} \in \{0,1\}$`: unit `$i$` treated at time `$t$`.

`$$\hat\beta^{DD} = \Delta \bar y_{\text{treat}} - \Delta \bar y_{\text{ctrl}}$$`

Unbiased for the ATT under parallel trends. No controversy.
]

By Frisch-Waugh-Lovell:

`$$\hat\beta^{DD} = \frac{\widehat{Cov}(y_{it}, \tilde D_{it})}{\widehat{Var}(\tilde D_{it})}$$`

where `$\tilde D_{it}$` is `$D_{it}$` residualized on both fixed effects.
]

.small[
.highlight-box[
Everything wrong with TWFE under staggered adoption hides inside
`$\tilde D_{it}$`: units treated *the whole panel* or *never* have no
variation left, and mid-panel switchers dominate the variance.
]
]

<a href="#fwl-detail" class="nav-btn">FWL detail</a>

---

# The Application: Staggered Zone-Notification Rollout

30 cities, 60 weeks. Cohorts adopt at week 15, 25, 35; six cities never do.
Outcome `$y$`: log completed trips (read: engagement or revenue per customer
in a retail setting).

.tiny[
Units here are cities, which keeps cross-unit interference plausibly small.
If units were sellers or customers inside one marketplace (shared
inventory, cannibalization, reallocation), spillovers would contaminate
not-yet-treated controls, and no staggered-DiD estimator fixes that; unit
choice is the first identification decision.
]

---

# The Same Problem at an Online Retailer

- A large online retailer upgrades metros from two-day to next-day delivery as new fulfillment centers open; the panel is metro-week orders, with three adoption cohorts and a slice of never-upgraded metros.
- TWFE on this panel uses already-upgraded metros as controls for late-upgrading metros during the late cohorts' post periods: the forbidden comparison.
- Delivery effects are dynamic: customers update their ordering behavior as they learn the reliability of the faster promise, so effects grow with time since adoption.
- The forbidden comparison subtracts the early cohort's growing effect from the late cohort's estimate, attenuating TWFE toward zero.
- Holding out a slice of metros that never receive next-day delivery preserves the clean comparison group that every estimator in this course relies on.

---

# Three Heterogeneity Scenarios, One DGP

Same panel structure; only the treatment-effect function changes.

.small[
| Scenario | Effect for cohort `$g$` at time `$t$` | Interpretation |
|---|---|---|
| `constant` | `$1.0$` | one-time level shift, same for everyone |
| `dynamic` | `$0.4\,\lbrack 1 + 0.10 (t-g)\rbrack$` | effect grows with exposure, same across cohorts |
| `heterogeneous` | `$0.4\,\lbrack 1 + 0.10 (t-g)\rbrack\,\lbrack 1 + 0.8\cdot\mathbb{1}(g{=}15) - 0.8\cdot\mathbb{1}(g{=}35)\rbrack$` | early adopters gain most |
]

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Scenario </th>
   <th style="text-align:right;"> True ATT </th>
   <th style="text-align:right;"> TWFE estimate </th>
   <th style="text-align:right;"> Bias (%) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> constant </td>
   <td style="text-align:right;"> 1.000 </td>
   <td style="text-align:right;"> 0.997 </td>
   <td style="text-align:right;"> -0.312 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dynamic </td>
   <td style="text-align:right;"> 1.137 </td>
   <td style="text-align:right;"> 0.686 </td>
   <td style="text-align:right;"> -39.651 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> heterogeneous </td>
   <td style="text-align:right;"> 1.407 </td>
   <td style="text-align:right;"> 0.423 </td>
   <td style="text-align:right;"> -69.904 </td>
  </tr>
</tbody>
</table>

.highlight-box[
TWFE recovers the constant effect, loses 40% of the dynamic one, and 70%
once cohorts differ. Nothing about the code changed. The estimator did this.
]

---

# Goodman-Bacon (2021): The Theorem

With `$K$` timing groups (adoption dates), the TWFE coefficient is **exactly**

`$$\hat\beta^{DD} = \sum_{k \neq U} s_{kU}\, \hat\beta_{kU} + \sum_{k} \sum_{l > k} \left[ s_{kl}^{k}\, \hat\beta_{kl}^{k} + s_{kl}^{l}\, \hat\beta_{kl}^{l} \right]$$`

a weighted average of **every possible 2x2 DiD** in the data, with weights
`$s \geq 0$` summing to 1.

The three species in the zoo:

.small[
| 2x2 | Treatment group | Control group | Window |
|---|---|---|---|
| `$\hat\beta_{kU}$` | cohort `$k$` | never-treated `$U$` | full panel |
| `$\hat\beta_{kl}^{k}$` | earlier cohort `$k$` | later cohort `$l$`, *not yet treated* | `$t < g_l$` |
| `$\hat\beta_{kl}^{l}$` | later cohort `$l$` | earlier cohort `$k$`, **already treated** | `$t \geq g_k$` |
]

.blue-box[
The first two are legitimate DiDs. The third uses already-treated units as
controls: the **forbidden comparison**.
]

<a href="#bacon-derivation" class="nav-btn">derivation</a>

---

# Who Gets Compared to Whom

---

# The Weights

Let `$n_k$` be group sizes, `$n_{kl} = n_k / (n_k + n_l)$` relative sizes, and
`$\bar D_k$` the share of the panel cohort `$k$` spends treated. Then:

`$$s_{kl}^{k} \propto \left[(n_k + n_l)(1 - \bar D_l)\right]^2 \; n_{kl}(1 - n_{kl}) \; \frac{\bar D_k - \bar D_l}{1 - \bar D_l} \cdot \frac{1 - \bar D_k}{1 - \bar D_l}$$`

`$$s_{kl}^{l} \propto \left[(n_k + n_l)\bar D_k\right]^2 \; n_{kl}(1 - n_{kl}) \; \frac{\bar D_l}{\bar D_k} \cdot \frac{\bar D_k - \bar D_l}{\bar D_k}$$`
]

**What drives a weight up:**

- Bigger subsamples: the `$(n_k + n_l)^2$` and `$n_{kl}(1-n_{kl})$` terms.
- Treatment-share variance `$\bar D(1-\bar D)$`: cohorts treated **near the
  middle of the panel** get the most weight. Units treated very early or
  very late contribute little identifying variance.

.highlight-box[
Weights come from the *design* (sizes and timing), not from effect
precision. TWFE decides what to estimate based on when your PM shipped.
]

<a href="#weights-detail" class="nav-btn">where these come from</a>

---

# The Forbidden Comparison, Up Close

Late cohort (g=35) as treatment, Early cohort (g=15) as control, window
`$t \geq 15$`. Both cohorts have strictly positive, growing effects.

The early cohort's *growing* effect is part of the control trend and gets
**subtracted**. This 2x2 estimates `$\text{ATT}_{late} - \Delta\text{ATT}_{early}$`,
not an ATT.

<a href="#forbidden-detail" class="nav-btn">the algebra</a>

---

# `bacondecomp::bacon()` in Practice

```r
library(bacondecomp)
df <- panel |> # bacon() wants a plain data.frame, numeric treatment
  transmute(city, t, y, treat = as.numeric(treated)) |>
  as.data.frame()
bd <- bacon(y ~ treat, data = df, id_var = "city", time_var = "t")
```

```r
bd |> group_by(type) |> # aggregate the 9 2x2s to the 3 species
  summarise(avg_estimate = weighted.mean(estimate, weight),
            weight = sum(weight)) |>
  mutate(across(where(is.numeric), ~ round(.x, 3)))
```

```
## # A tibble: 3 × 3
##   type                     avg_estimate weight
##   <chr>                           <dbl>  <dbl>
## 1 Earlier vs Later Treated        0.637  0.184
## 2 Later vs Earlier Treated        0.079  0.317
## 3 Treated vs Untreated            1.09   0.499
```

.highlight-box[
A third of the total weight sits on forbidden comparisons averaging 0.08,
against legitimate comparisons averaging around 1.1. That is the whole
attenuation story in one table.
]

---

# The Diagnostic Plot: Weight vs Estimate

Every applied seminar wants this plot before believing a staggered DiD.

---

# The Decomposition Is Exact

```r
c(weighted_sum = sum(bd$estimate * bd$weight),
  twfe         = coef(feols(y ~ treated | city + t, data = panel))[[1]])
```

```
## weighted_sum         twfe 
##    0.6861924    0.6861924
```

Identical to machine precision. This is an algebraic identity about OLS,
**not an approximation and not an estimator of anything new**.

`$$\text{plim}\;\hat\beta^{DD} = \text{VWATT} - \Delta\text{ATT}$$`

a *variance-weighted* ATT minus a term collecting **within-cohort effect
changes** that forbidden comparisons subtract. Constant effects kill
`$\Delta\text{ATT}$`; dynamics feed it.
]

<a href="#estimand-detail" class="nav-btn">full estimand decomposition</a>

---

# When TWFE Is Perfectly Fine

Run the identical decomposition on the `constant` scenario:

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> type </th>
   <th style="text-align:right;"> avg_estimate </th>
   <th style="text-align:right;"> weight </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Earlier vs Later Treated </td>
   <td style="text-align:right;"> 0.972 </td>
   <td style="text-align:right;"> 0.184 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Later vs Earlier Treated </td>
   <td style="text-align:right;"> 0.990 </td>
   <td style="text-align:right;"> 0.317 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Treated vs Untreated </td>
   <td style="text-align:right;"> 1.010 </td>
   <td style="text-align:right;"> 0.499 </td>
  </tr>
</tbody>
</table>

Weights are unchanged (they depend only on the design). But now **every 2x2
estimates the same 1.0**, so the weighting is harmless.

**TWFE is unbiased for the ATT when:**

1. Effects are homogeneous across cohorts *and* constant in event time; or
2. Adoption is *not* staggered (single treatment date, 2x2 world); or
3. You are willing to interpret VWATT as the estimand and effects are
   stable over time.

.highlight-box[
The interview answer is not "TWFE is broken". It is: "TWFE estimates a
variance-weighted average of 2x2s; here is when that coincides with the
ATT, and here is the diagnostic I would run."
]

---

# From Attenuation to Sign Reversal

Bacon weights are non-negative *on the 2x2 estimates*. The scary headlines
("negative weights") are about the implied weights **on the underlying
cohort-time effects** `$ATT(g,t)$` inside those 2x2s.

- Each forbidden `$\hat\beta_{kl}^{l}$` contains `$-\Delta ATT_k$`: early-cohort
  effect *growth* enters with a negative sign.
- Our panel resists a full flip because 6 never-treated cities anchor half
  the weight on clean comparisons. **Remove them** (everyone eventually
  adopts, common at a company that ships everywhere) and forbidden 2x2s take
  over.

```r
set.seed(42) # no never-treated group: 3 cohorts of 10, everyone adopts
flip <- expand_grid(city = 1:30, t = 1:60) |>
  left_join(tibble(city = 1:30, g = rep(c(15, 25, 35), each = 10)), by = "city") |>
  mutate(treated = t >= g,
         eff = if_else(treated, 0.2 + 0.15 * (t - g), 0), # mild dynamics
         y = 5 + 0.05 * t + city * 0.1 + eff + rnorm(n(), 0, 0.3))
c(true_ATT = mean(flip$eff[flip$treated]),
  TWFE     = coef(feols(y ~ treated | city + t, data = flip))[[1]])
```

```
##   true_ATT       TWFE 
##  2.9638889 -0.1448608
```

Every `$ATT(g,t) > 0$`, TWFE **negative**. de Chaisemartin and D'Haultfoeuille
(2020) formalize the implicit weights behind this; that is Module 2
territory.

---

# Design the Rollout: Reserve a Never-Treated Slice

The sign-reversal demo above is not just a diagnosis, it is an argument you
can make **before** a staggered rollout ships. If you are consulted on the
design, argue to hold out a slice of markets or cohorts, the panel analogue
of a global holdout in an A/B test.

- A platform-wide launch cannot be a customer-level experiment, but **which
  cities wait** is still a design choice you can make.
- Even a small never-treated set restores clean comparisons for every
  estimator in this course, not just TWFE: it anchors weight away from
  forbidden 2x2s here, and it is the control group Callaway-Sant'Anna and
  BJS need in Module 2.
- The quantified case is the demo above: with 6 never-treated cities the
  panel resists a full sign flip; remove them and it does not.

.blue-box[
"Ship everywhere" is an ops default, not a technical requirement. A held-out
slice costs some short-run coverage and buys identification for the entire
toolbox that follows.
]

---

# Practitioner Checklist for Any Staggered Rollout

Before reporting a staggered TWFE estimate:

.small[
| Step | Tool | What you learn |
|---|---|---|
| 1. Map the cohorts | `count(g)` | how much never-treated glue you have |
| 2. Bacon decomposition | `bacondecomp::bacon()` | weight on forbidden comparisons |
| 3. Weight-vs-estimate scatter | previous slide | which 2x2s drive the coefficient |
| 4. Compare vs robust estimator | CS / SA / BJS (Module 2) | how much the answer moves |
| 5. Event-study with robust estimator | Module 2 | dynamics, pre-trends |
]

.blue-box[
**Rules of thumb.** Forbidden weight near zero and effects plausibly stable:
TWFE is probably fine, say why. Forbidden weight 30%+ or visible dynamics:
robust estimators are not optional.
]

At platform scale (staggered feature launches across marketplaces, fulfillment
centers, product categories), staggering is the *default*, not the edge case.

---

# Interview Questions

.small[
| Question | Core of a strong answer |
|---|---|
| "We rolled a feature out city by city and ran TWFE. What could go wrong?" | Staggered timing makes TWFE a weighted average of 2x2s including already-treated controls; with dynamic effects it is biased toward zero or beyond. Run the Bacon decomposition. |
| "What exactly is a forbidden comparison?" | A 2x2 using an earlier-treated cohort as control during its own post period; its effect *changes* get subtracted from the later cohort's estimate. |
| "When would you still trust TWFE?" | Non-staggered designs; effects constant in event time and across cohorts; or when forbidden-comparison weight is negligible and you can defend effect stability. |
| "Can TWFE flip the sign of a true positive effect?" | Yes: strong effect dynamics make forbidden 2x2s increasingly negative, and they can dominate. Demonstrated with all `$ATT(g,t) > 0$` and TWFE `$< 0$`. |
| "What determines the Bacon weights?" | Group sizes and treatment-share variance `$\bar D(1-\bar D)$`: mid-panel cohorts dominate. Design, not effect size or precision. |
| "Ops staggered the launch so every city eventually got it. That is even better than a holdout, right?" | No: with no never-treated slice, late cohorts get compared against already-treated cities, and TWFE can even flip the sign of a true positive effect (this deck's demo). Reserve a never-treated holdout before the launch ships. |
]

---

# Going Deeper

.small[
| Paper | What it adds |
|---|---|
| Goodman-Bacon (2021), *J. Econometrics* | The decomposition on these slides: exact weights, VWATT vs `$\Delta$`ATT estimand reading. |
| de Chaisemartin and D'Haultfoeuille (2020), *AER* | Implicit ATT(g,t) weights for general two-way FE designs; can be negative; robust switcher estimator. |
| Borusyak, Jaravel and Spiess (2024), *ReStud* | Imputation view: efficiency under parallel trends, the "forbidden extrapolation" framing. |
| Baker, Larcker and Wang (2022), *JFE* | How much this matters in published finance DiDs: a lot. |
| Roth, Sant'Anna, Bilinski and Poe (2023), *J. Econometrics* | Survey tying the whole modern DiD literature together. |
]

**Next module:** the estimators that fix this: Callaway-Sant'Anna,
Sun-Abraham, BJS imputation, dCDH.

**Drill:** `exercise.R` hand-codes all nine 2x2s and their weights, and
replicates `bacon()` to 1e-14.

---

# Backup Slides

---

# Backup: The DGP

```r
make_panel <- function(scenario = c("constant", "dynamic", "heterogeneous"),
                       n_cities = 30, n_t = 60, seed = 42) {
  scenario <- match.arg(scenario)
  set.seed(seed)
  cohorts <- tibble(
    city = 1:n_cities,
    g = c(rep(15, 8), rep(25, 8), rep(35, 8), rep(Inf, 6)) # adoption week
  )
  expand_grid(city = 1:n_cities, t = 1:n_t) |>
    left_join(cohorts, by = "city") |>
    mutate(
      treated = t >= g,
      eff = case_when(
        !treated ~ 0,
        scenario == "constant"      ~ 1.0,
        scenario == "dynamic"       ~ 0.4 * (1 + 0.10 * (t - g)),
        scenario == "heterogeneous" ~ 0.4 * (1 + 0.10 * (t - g)) *
                                      (1 + 0.8 * (g == 15) - 0.8 * (g == 35))
      ),
      y = 5 + 0.05 * t + city * 0.1 + eff + rnorm(n(), 0, 0.3) # common trend
    )
}
```
]

.small[
- City FE ( `$0.1 \times$` city index) and a common weekly trend `$0.05t$`: parallel
  trends holds **by construction**. Every bias on the main slides is purely the
  estimator, never a PT violation.
- Noise `$\sigma = 0.3$` is small relative to effects, so decomposition tables are
  stable and readable.
]

---

# Backup: The FWL View of TWFE

`$$\tilde D_{it} = D_{it} - \bar D_i - \bar D_t + \bar{\bar D}$$`

`$$\hat\beta^{DD} = \frac{\sum_{it} \tilde D_{it}\, y_{it}}{\sum_{it} \tilde D_{it}^2}$$`

Three consequences:

1. `$y$` enters linearly with weights `$\tilde D_{it} / \sum \tilde D^2$`: TWFE is a
   fixed linear combination of cell means, which is why an exact decomposition
   into 2x2s exists at all.
2. A unit treated for the entire panel has `$D_{it} = \bar D_i = 1$`, so
   `$\tilde D_{it} \approx 0$`: always-treated units contribute *nothing* to
   identification, they only sharpen the FE estimates.
3. `$\sum \tilde D_{it}^2$` is maximized for units treated half the panel: this is
   exactly the `$\bar D_k (1 - \bar D_k)$` term in the Bacon weights.

The decomposition then follows from partitioning `$\sum \tilde D \tilde y$` by
timing-group pairs and completing each piece into a 2x2 DiD (next backup).
]

---

# Backup: Deriving the Decomposition (1 of 2)

.small[
Setup: timing groups `$a$` (earlier, adoption `$g_a$`), `$b$` (later), `$U$` (never).
Within any **pair** of groups, restricted to the right **window**, the panel is
a plain 2x2: one group changes status, the other does not.

**Step 1: partition the identifying variance.** Write the total demeaned
variance `$\widehat{Var}(\tilde D)$` as a sum over group pairs and windows.
Every `$(i,t)$` cell belongs to some pair-window combination, and within each,
demeaned treatment has the 2x2 structure.

**Step 2: each piece is a 2x2 DiD.** For a pair `$(k, U)$`, OLS on the subsample
is textbook DiD:

`$$\hat\beta_{kU} = \left[\bar y_k^{post(k)} - \bar y_k^{pre(k)}\right] -
\left[\bar y_U^{post(k)} - \bar y_U^{pre(k)}\right]$$`

For timing pairs `$(k, l)$` with `$g_k < g_l$`, the window splits in two:

- `$t < g_l$`: only `$k$` switches. Control = `$l$` (not yet treated). This is
  `$\hat\beta_{kl}^{k}$`.
- `$t \geq g_k$`: only `$l$` switches. Control = `$k$` (treated throughout). This is
  `$\hat\beta_{kl}^{l}$`, the forbidden one.
]

---

# Backup: Deriving the Decomposition (2 of 2)

.small[
**Step 3: reassemble.** OLS on the full sample equals the variance-weighted
average of the subsample OLS coefficients:

`$$\hat\beta^{DD} = \sum_{\text{pieces } p} \underbrace{\frac{\widehat{Var}_p(\tilde D)\, n_p}{\sum_q \widehat{Var}_q(\tilde D)\, n_q}}_{s_p} \hat\beta_p$$`

Computing `$\widehat{Var}_p(\tilde D)$` for each piece type gives the formulas on
the weights slide:

- Within a `$(k,U)$` piece, treatment share is `$n_{kU}$` across units and
  `$\bar D_k$` across time: variance `$n_{kU}(1-n_{kU}) \bar D_k(1-\bar D_k)$`.
- The timing pieces get the analogous terms with the window lengths
  `$(1 - \bar D_l)$` or `$\bar D_k$` scaling how much panel they occupy: the
  squared leading factors.

**Why weights are positive:** they are variances times sample shares. All the
sign trouble lives inside the `$\hat\beta_{kl}^{l}$` *estimates*, not the weights.

Full details: Goodman-Bacon (2021), Theorem 1 and Appendix A. The exercise
implements exactly these formulas and matches `bacon()` to 1e-16 on weights.
]

---

# Backup: Reading the Weight Formulas

Three factors, three design questions:

| Factor | Question it answers |
|---|---|
| `$(n_k + n_U)^2$` | how big is this subsample relative to the panel? |
| `$n_{kU}(1 - n_{kU})$` | how balanced are treatment and control *groups*? |
| `$\bar D_k (1 - \bar D_k)$` | how balanced are pre and post *periods*? |

For timing pairs, the same three factors appear, but the window is shorter:
the leading factor becomes `$\left[(n_k+n_l)(1-\bar D_l)\right]^2$` for the
early-vs-late piece (window is `$l$`'s pre-period, length share `$1-\bar D_l$`)
and `$\left[(n_k+n_l)\bar D_k\right]^2$` for the forbidden piece (window is
`$k$`'s post-period, length share `$\bar D_k$`).

**Worked example from the deck's panel:** cohort g=25 spends
`$\bar D = 36/60 = 0.6$` of the panel treated, closest of the three cohorts to
`$0.5$`, and the never-treated group is large, so `$s_{25,U} = 0.18$` is the
biggest single vs-never weight even though all cohorts have 8 cities.

Consequence worth saying in an interview: **two rollouts with identical
effects but different calendars give different TWFE estimands.**
]

---

# Backup: The Forbidden 2x2, Algebraically

.small[
Take the pair early `$k$` (adopts `$g_k$`), late `$l$` (adopts `$g_l > g_k$`), window
`$t \geq g_k$` where `$k$` is treated throughout. Split the window at `$g_l$`:
`$W_1 = \lbrack g_k, g_l)$` and `$W_2 = \lbrack g_l, T\rbrack$`. With parallel
trends in untreated outcomes and writing `$\text{ATT}_k(W)$` for cohort `$k$`'s
average effect over window `$W$`:

`$$\hat\beta_{kl}^{l} \;\xrightarrow{p}\;
\underbrace{\text{ATT}_l(W_2)}_{\text{what you want}} \;-\;
\underbrace{\left[\text{ATT}_k(W_2) - \text{ATT}_k(W_1)\right]}_{\text{control group's effect drift}}$$`

- **Constant effects:** `$\text{ATT}_k(W_2) = \text{ATT}_k(W_1)$`, the drift term
  dies, and the forbidden 2x2 is a fine estimate of `$\text{ATT}_l$`.
- **Dynamic effects:** `$k$`'s effect keeps growing into `$W_2$`, the drift term is
  positive and gets subtracted: attenuation, and with strong dynamics, sign
  reversal.

In the deck's dynamic panel: the (35 vs 15) forbidden estimate is `$-0.005$`
while the true `$\text{ATT}_{35}(W_2) \approx 0.9$`. The early cohort's drift
over weeks 35-60 was that large.

This is also why the *constant* scenario is immune despite identical weights:
weights did not change, the drift terms all vanished.
]

---

# Backup: The Estimand Decomposition

`$$\text{plim}\; \hat\beta^{DD} = \text{VWATT} + \text{VWCT} - \Delta\text{ATT}$$`

| Term | Meaning | Dies when |
|---|---|---|
| VWATT | variance-weighted ATT: each 2x2's ATT weighted by the Bacon `$s$` | never (this is the "good" part) |
| VWCT | variance-weighted common trend: PT violations across pairs | parallel trends holds |
| `$\Delta$`ATT | weighted sum of *within-cohort effect changes* subtracted by forbidden 2x2s | effects constant in event time |

Three readings:

1. Even with PT and constant effects, TWFE targets **VWATT**, not the ATT: the
   variance weighting is a different aggregation than "average over treated
   cells". With constant effects they coincide.
2. Dynamics enter *only* through `$\Delta$`ATT, and always with a minus sign:
   dynamic effects bias TWFE **toward zero and beyond**, never away from zero.
3. PT violations and heterogeneity bias are *separate terms*: fixing one does
   not fix the other. Module 2 fixes `$\Delta$`ATT; Module 3 stress-tests VWCT.
]