Module 2: Heterogeneity-Robust DiD

.title[
# Module 2: Heterogeneity-Robust DiD
]
.subtitle[
## Callaway-Sant’Anna, Sun-Abraham, Imputation, and Switchers
]

---

# Course Map

<table>
<tr><th>#</th><th>Module</th><th>Status</th></tr>
<tr><td>1</td><td><a href="../module-01/slides.html">TWFE Diagnosed: Goodman-Bacon and the Zoo of 2x2s</a></td><td>done</td></tr>
<tr><td><b>2</b></td><td><b>Heterogeneity-Robust DiD: CS, SA, BJS, dCDH</b> <i>(you are here)</i></td><td>current</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

Module 1 showed *why* TWFE breaks under staggered adoption: it averages in
forbidden 2x2s that use already-treated units as controls. This module is the
repair kit, at the level of detail an interview will probe.

**You will be able to:**

1. Define the group-time ATT and identify it with never-treated vs
   not-yet-treated controls, universal vs varying base period.
2. Explain the doubly-robust estimator and CS's influence-function SEs and
   uniform bands.
3. Diagnose why a naive TWFE event study shows pre-trends even when parallel
   trends holds, and fix it with Sun-Abraham.
4. Say what BJS imputation and dCDH switchers add, and choose among all four.
5. Pick the right aggregation (simple / group / dynamic) for the question.

**Same application as Module 1:** staggered zone-notification rollout, 30
cities, 60 weeks, cohorts at 15/25/35, 6 never-treated, parallel trends true
by construction. All numbers are the `heterogeneous` scenario. Outcome:
log completed trips (read: engagement or revenue per customer in a retail
setting). The rollout itself is ops-driven, not a designed experiment;
these estimators are what you reach for when the timing was chosen for
business reasons, not randomized.

---

# The Same Problem at an Online Retailer

- Cohort `$g$` = the set of metros upgraded to next-day delivery in quarter `$g$`; `$\text{ATT}(g,t)$` measures the delivery effect for that cohort at calendar quarter `$t$`.
- Event-time aggregation of `$\text{ATT}(g,t)$` traces the habit-formation path: a rising event-study plot confirms that effects grow with months since the upgrade.
- The preferred control group is never-upgraded metros (still on two-day delivery); not-yet-upgraded metros are a valid but noisier alternative when the never-treated slice is small.
- Cross-cohort heterogeneity (early metros differ in demand density from late metros) is captured naturally by the ATT(g,t) surface rather than collapsed into a single TWFE number.
- Callaway-Sant'Anna or BJS avoids the forbidden comparison that biases TWFE in this panel, giving a clean estimate for each cohort-quarter cell.

---

# What a Fix Must Do

Every heterogeneity-robust estimator obeys one rule: **only clean
comparisons**. Never use an already-treated unit as a control.

.small[
| Estimator | Core idea | Package | This module adds |
|---|---|---|---|
| **Callaway-Sant'Anna** | group-time `$\text{ATT}(g,t)$`, then aggregate | `did` | DR estimator, IF-based SEs, uniform bands |
| **Sun-Abraham** | saturated cohort `$\times$` event-time regression | `fixest` | the contamination algebra, when it equals CS |
| **Borusyak-Jaravel-Spiess** | impute `$Y(0)$` from untreated cells, average | `didimputation` | efficiency under parallel trends |
| **de Chaisemartin-D'Haultfoeuille** | switchers vs stayers, negative-weights audit | `DIDmultiplegt` | the switcher estimand, hand-coded |
]

.blue-box[
They agree in simple cases and diverge under heterogeneity. The interview
question is never "which package" but "which estimand, which controls, which
aggregation, and how would you check the answer moved for the right reason".
]

---

# Callaway-Sant'Anna: The Estimand

TWFE forced a two-dimensional object into one number. CS keeps both
dimensions. The building block is the **group-time ATT**:

`$$\text{ATT}(g, t) = \mathbb{E}\!\left[Y_t(g) - Y_t(0) \mid G = g\right]$$`

the effect at calendar time `$t$` on the cohort first treated at `$g$`.

**Time index `$t$`**: when we evaluate the effect.

**Event time** `$e = t - g$`: exposure length, aligns cohorts at `$e = 0$`.
]

.pull-right[
.highlight-box[
Two indices, never collapsed prematurely. A headline number is a *choice* of
how to average this surface, made explicit and last, not baked into the
estimator like TWFE's variance weights.
]
]

For `$t \geq g$`, under parallel trends and no anticipation, each
`$\text{ATT}(g,t)$` is a clean 2x2 DiD anchored at a pre-period base.

---

# Two Identification Levers

`$$\text{ATT}(g, t) = \big[\bar Y_{g,t} - \bar Y_{g,g-1}\big] - \big[\bar Y_{C,t} - \bar Y_{C,g-1}\big]$$`

**Comparison group `$C$`:**

- **Never-treated**: only the 6 cities that never adopt. Cleanest assumption,
  but high variance when that group is small.
- **Not-yet-treated**: for each `$(g,t)$`, add every cohort with `$G > t$` (plus
  never-treated). More controls, lower variance, same parallel-trends logic
  extended to the not-yet-treated.

**Base period:**

- **Universal**: anchor at `$g-1$` for all `$t$`. Each `$\text{ATT}(g,t)$` is a
  direct "did the gap move from `$g-1$` to `$t$`?" test. Makes CS coincide with
  Sun-Abraham.
- **Varying**: anchor at `$t-1$`, re-anchoring as `$t$` moves. Natural for long
  differences, but the reference drifts.

.blue-box[
Here: never-treated + universal base gives `$\text{ATT}(15,\cdot)$` high,
`$\text{ATT}(35,\cdot)$` near zero, the cohort heterogeneity TWFE smeared into
0.42. <a href="#cs-dr-derivation" class="inline-btn">doubly-robust form</a>
]

---

# CS in Practice and the ATT(g,t) Surface

```r
dat <- panel |>            # did wants 0 = never
  mutate(g0 = ifelse(is.finite(g), g, 0)) |>
  select(city, t, y, g0) |>
  as.data.frame()

cs <- att_gt(
  yname = "y", tname = "t",
  idname = "city", gname = "g0",
  data = dat,
  control_group = "nevertreated",
  base_period = "universal",
  est_method = "dr")          # doubly robust
```

Each cohort's effect grows with exposure and the early cohort dwarfs the late
one. This 2D structure is exactly what one TWFE number destroys.
]
]

.pull-right[
<img src="slides_files/figure-html/cs-plot-1.png" style="display: block; margin: auto;" />
]

---

# Inference: Influence Functions, Uniform Bands

CS derives the **influence function** of every `$\widehat{\text{ATT}}(g,t)$`:
the per-observation score whose sample variance is the estimator's variance,
accounting for the estimated nuisance models.

Because the influence functions for *all* `$(g,t)$` are available jointly, a
**multiplier bootstrap** gives simultaneous (uniform) confidence bands over
the whole ATT path, not one point at a time.

```r
c(pointwise_crit = 1.96,
  uniform_crit   = round(cs_dynamic$crit.val.egt, 2))  # dynamic aggregation
```

```
##   pointwise_crit uniform_crit.95% 
##             1.96             2.80
```

.highlight-box[
The uniform critical value (2.8 here) exceeds
1.96, so honest event-study plots use the wider band. `did` clusters by unit
and turns on `bstrap` + `cband` by default. Reporting pointwise intervals on a
12-point event study invites a multiple-comparisons objection you will not
survive in a review.
]

---

# Aggregation: Three Questions, Three Answers

`$\text{ATT}(g,t)$` is a surface. The headline is a weighted average, and the
weights are a modeling choice.

.small[
| Aggregation | Averages over | Answers |
|---|---|---|
| **simple** | every treated `$(g,t)$` cell, equal weight | average lift across all treated place-weeks |
| **group** | cohorts (by size), each cohort's mean post-effect | which cohorts drove it |
| **dynamic** | event time `$e$`, equal weight per `$e$` | is the effect still growing with exposure |
]

```r
aggte(cs, type = "simple")    # one number, cell-weighted
aggte(cs, type = "group")     # per-cohort, then average
aggte(cs, type = "dynamic")   # event-study path + overall
```

.blue-box[
Match the aggregation to the decision. "Did the rollout help on average?" is
*simple*. "Should we prioritize early markets next time?" is *group*. "Has the
effect plateaued?" is *dynamic*.
]

---

# They Differ Numerically, By Construction

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Aggregation </th>
   <th style="text-align:right;"> True target </th>
   <th style="text-align:right;"> CS estimate </th>
   <th style="text-align:right;"> Std. error </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> simple </td>
   <td style="text-align:right;"> 1.407 </td>
   <td style="text-align:right;"> 1.661 </td>
   <td style="text-align:right;"> 0.245 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> group </td>
   <td style="text-align:right;"> 1.207 </td>
   <td style="text-align:right;"> 1.408 </td>
   <td style="text-align:right;"> 0.251 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dynamic </td>
   <td style="text-align:right;"> 1.792 </td>
   <td style="text-align:right;"> 2.090 </td>
   <td style="text-align:right;"> 0.159 </td>
  </tr>
</tbody>
</table>

Three different estimands: true targets 1.41, 1.21, 1.79. Dynamic is largest
because it equal-weights event times, and at long exposure only the early
high-effect cohort survives. Group is smallest because the low-effect late
cohort gets equal cohort weight.

.highlight-box[
The CS estimates sit above their targets by a fixed amount here. That offset
is *identical across all three effect scenarios*: the signature of the small
6-city never-treated draw, not bias.
<a href="#agg-offset-detail" class="nav-btn">why it is a draw, not bias</a>
]

---

# Sun-Abraham: The Contamination Problem

The reflex diagnostic is the TWFE event study:

`$$y_{it} = \alpha_i + \lambda_t + \sum_{e \neq -1} \mu_e\,
\mathbb{1}\{t - g_i = e\} + \varepsilon_{it}$$`

and you read `$\mu_e$` as "the effect `$e$` periods after adoption".

Sun-Abraham: `$\mu_e$` is **not** the average effect at event time `$e$`. It is a
weighted sum of cohort effects at `$e$` *and at other event times*, with weights
that can be negative and need not sum to one within `$e$`.

.highlight-box[
Consequence: a **pre-period** coefficient `$\mu_{-3}$` can pick up another
cohort's **post**-treatment effect. You get a nonzero pre-trend even when
parallel trends holds exactly. The pretrend test fails for a reason that has
nothing to do with pretrends.
<a href="#sa-contam-derivation" class="nav-btn">the weighting</a>
]

---

# Contamination, With the Noise Switched Off

Same panel, deterministic (drop the noise term). Parallel trends is now exact,
so any nonzero pre-coefficient is *pure contamination*.

Naive TWFE (red) has nonzero leads and attenuated lags. SA and CS sit exactly
on Truth. <a href="#es-noisy-detail" class="inline-btn">with noise</a>

---

# Sun-Abraham: The Interaction-Weighted Fix

Saturate the regression: interact **every cohort** with **every event time**,

`$$y_{it} = \alpha_i + \lambda_t + \sum_{g}\sum_{e \neq -1}
\delta_{g,e}\,\mathbb{1}\{G_i = g\}\,\mathbb{1}\{t - g = e\} + \varepsilon_{it}$$`

so `$\delta_{g,e}$` is a clean cohort-specific effect, then average across
cohorts at each `$e$` using **cohort shares** as weights (non-negative, sum to
one).

```r
panel_sa <- panel |> mutate(g_sa = ifelse(is.finite(g), g, 10000))  # never -> big
fit_sa <- feols(y ~ sunab(g_sa, t) | city + t, data = panel_sa)
summary(fit_sa, agg = "att")   # cohort-share-weighted overall ATT
```

.blue-box[
`sunab()` builds the saturated design and the aggregation in one call. The
never-treated cohort is coded to a large finite value so it is the excluded,
uncontaminated control. No leakage across event times: that is the whole
repair.
]

---

# When Sun-Abraham Equals Callaway-Sant'Anna

With **never-treated controls** and the **same base period** `$g-1$`, the IW
estimator and CS are the same clean comparison.

```r
c(sun_abraham   = round(sa_att[["Estimate"]], 5),
  cs_nev_simple = round(cs_simple$overall.att, 5),
  difference    = signif(abs(sa_att[["Estimate"]] - cs_simple$overall.att), 3))
```

```
##   sun_abraham cs_nev_simple    difference 
##   1.66061e+00   1.66061e+00   1.24000e-11
```

Identical to 1e-11, and the two event-study paths match at every `$e$`.

.highlight-box[
So Sun-Abraham is not an *alternative* to CS: it is CS in regression clothing.
The standard errors differ (SA 0.098, CS
0.245) because each estimates the control-group
sampling variance its own way, but the point estimate is the same object. Use
whichever fits your workflow; do not report both as a robustness check.
]

---

# Borusyak-Jaravel-Spiess: Imputation

The most efficient of the four, and the most transparent. Three steps:

1. Fit `$Y_{it} = \alpha_i + \lambda_t + \varepsilon_{it}$` on **untreated cells
   only** (all not-yet-treated and never-treated observations, including
   treated units' pre-periods).
2. Impute `$\hat Y_{it}(0) = \hat\alpha_i + \hat\lambda_t$` for every treated
   cell.
3. Effect `$= Y_{it} - \hat Y_{it}(0)$`; average with your target's weights.

```r
did_imputation(dat, yname = "y", gname = "g0", tname = "t", idname = "city")
```

.blue-box[
The mirror image of Module 1's forbidden comparison: **never extrapolate a
treated cell's counterfactual from other treated cells**, only from untreated
ones. Step 1 is exactly that discipline.
<a href="#bjs-algorithm-detail" class="nav-btn">efficiency argument</a>
]

---

# Why Imputation Is More Efficient

Step 1 uses *every* untreated cell to pin the fixed effects, not just a 6-city
never-treated group. Lower-variance counterfactual, lower-variance estimate.

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Estimator </th>
   <th style="text-align:right;"> Estimate </th>
   <th style="text-align:right;"> Std. error </th>
   <th style="text-align:right;"> True target </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> BJS imputation </td>
   <td style="text-align:right;"> 1.406 </td>
   <td style="text-align:right;"> 0.033 </td>
   <td style="text-align:right;"> 1.407 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> CS (simple) </td>
   <td style="text-align:right;"> 1.661 </td>
   <td style="text-align:right;"> 0.245 </td>
   <td style="text-align:right;"> 1.407 </td>
  </tr>
</tbody>
</table>

BJS lands on the true simple ATT (1.407) with a standard
error of 0.033, roughly
7.3x tighter than CS on the
same panel.

.highlight-box[
The cost is the flip side: imputation leans entirely on the untreated model's
functional form, with no propensity-model insurance. If parallel trends is
shaky, use CS's doubly-robust option or stress-test with Module 3's Honest
DiD. Efficiency and robustness trade off.
]

---

# de Chaisemartin-D'Haultfoeuille: Switchers

Attack the problem one *switch* at a time. `$\text{DID}_M$` averages, over every
period where treatment status changes, the switchers' outcome change minus the
stayers' outcome change:

`$$\widehat{\text{DID}}_M = \sum_{t} \frac{N_t^{sw}}{\sum_s N_s^{sw}}
\left[\overline{\Delta Y}^{\,sw}_t - \overline{\Delta Y}^{\,stay\,0}_t\right]$$`

targeting the average **instantaneous** (event-time-zero) effect.

.small[
`DIDmultiplegt` needs X11 and does not load here, so hand-code the
instantaneous version and validate on the `dynamic` scenario (switch effect
0.4 by construction):
]

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> switch time </th>
   <th style="text-align:right;"> n switchers </th>
   <th style="text-align:right;"> DID_M </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> 15 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 0.732 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 25 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 0.305 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> 35 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 0.239 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> overall </td>
   <td style="text-align:right;"> 24 </td>
   <td style="text-align:right;"> 0.425 </td>
  </tr>
</tbody>
</table>

Overall 0.425 vs true 0.400. <a href="#dcdh-detail" class="inline-btn">negative weights</a>

---

# Four Estimators, One Table

Heterogeneous scenario, each estimate against the target it actually estimates.

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Estimator </th>
   <th style="text-align:right;"> Estimate </th>
   <th style="text-align:right;"> True target </th>
   <th style="text-align:left;"> Aggregation </th>
   <th style="text-align:right;"> Gap </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> TWFE (biased) </td>
   <td style="text-align:right;"> 0.423 </td>
   <td style="text-align:right;"> 1.407 </td>
   <td style="text-align:left;"> simple </td>
   <td style="text-align:right;"> -0.983 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> CS simple </td>
   <td style="text-align:right;"> 1.661 </td>
   <td style="text-align:right;"> 1.407 </td>
   <td style="text-align:left;"> simple </td>
   <td style="text-align:right;"> 0.254 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Sun-Abraham </td>
   <td style="text-align:right;"> 1.661 </td>
   <td style="text-align:right;"> 1.407 </td>
   <td style="text-align:left;"> simple </td>
   <td style="text-align:right;"> 0.254 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> BJS imputation </td>
   <td style="text-align:right;"> 1.406 </td>
   <td style="text-align:right;"> 1.407 </td>
   <td style="text-align:left;"> simple </td>
   <td style="text-align:right;"> 0.000 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> CS group </td>
   <td style="text-align:right;"> 1.408 </td>
   <td style="text-align:right;"> 1.207 </td>
   <td style="text-align:left;"> group </td>
   <td style="text-align:right;"> 0.202 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> CS dynamic </td>
   <td style="text-align:right;"> 2.090 </td>
   <td style="text-align:right;"> 1.792 </td>
   <td style="text-align:left;"> dynamic </td>
   <td style="text-align:right;"> 0.298 </td>
  </tr>
</tbody>
</table>

.highlight-box[
TWFE misses its target by -0.98 (attenuated toward
zero). The four robust estimators cluster near their targets; they differ from
each other because they answer different aggregation questions, not because
one is wrong. BJS is closest and tightest; CS/SA carry the never-treated draw.
]

---

# Choosing Among the Four

.small[
| Reach for | when | watch out for |
|---|---|---|
| **Callaway-Sant'Anna** (`did`) | you want group-time ATTs + flexible aggregation + DR robustness | high variance with a small never-treated group |
| **Sun-Abraham** (`fixest`) | you live in a regression workflow, want a clean event study | equals CS, not an independent robustness check |
| **BJS** (`didimputation`) | parallel trends is credible, efficiency matters | leans on the untreated model, no DR insurance |
| **dCDH** (`DIDmultiplegt`) | treatment turns on and off; you want the negative-weights audit | instantaneous estimand differs from a full dynamic ATT |
]

.blue-box[
**The workflow:** pick one workhorse (CS or BJS), report its event study with a
*uniform* band, robustness-check with a genuinely different estimator (CS vs
BJS, not SA vs CS). Agreement means the staggering is handled. Divergence means
parallel trends or the control group is doing the work: investigate before you
report.
]

---

# Interview Questions

.small[
| Question | Core of a strong answer |
|---|---|
| "You showed me a staggered DiD. Which robust estimator, and why?" | Name one (CS or BJS), state the estimand `$\text{ATT}(g,t)$` plus an explicit aggregation, the control group (never vs not-yet), and how you would check it moved for the right reason. Not "I ran the package". |
| "Your event study has a significant pre-trend but you say PT holds. Explain." | If it is a naive TWFE event study, the pre-coefficient is contaminated by other cohorts' post-effects (Sun-Abraham). Re-estimate with `sunab` or CS; a clean event study on the same data shows flat leads. |
| "CS and Sun-Abraham gave the same number. Is that a robustness check?" | No. With never-treated controls and base period `$g-1$` they are algebraically identical. A real check is CS vs BJS, which use different identification (2x2 vs imputation). |
| "Why might BJS beat CS on precision, and when would you not trust it?" | BJS pins the counterfactual with every untreated cell, so lower variance (here 7x). But it has no propensity-model insurance: if PT is shaky, the imputed `$Y(0)$` inherits the misspecification. Use DR-CS or Honest DiD. |
| "Simple, group, and dynamic aggregations disagree. Which is right?" | All are correct estimands of different questions. Pick the one matching the decision: average lift (simple), which cohorts (group), is it still growing (dynamic). Report which and why. |
| "You ran CS and imputation (BJS) and they diverge. What now?" | The divergence localizes the problem: CS and BJS lean on parallel trends and the control group differently, so a gap means one of those is doing more work than you think. Investigate pre-trends and control-group choice before reporting either estimate. |
| "Your four estimators give four numbers. Which one goes in the launch review?" | They answer different questions (cell-average vs cohort-average vs event path), not a menu to pick the best-looking one. Name the estimand the decision needs first, then report that estimator; if they diverge materially, say so, it is a diagnostic. |
]

---

# Going Deeper

.small[
| Paper | What it adds |
|---|---|
| Callaway and Sant'Anna (2021), *J. Econometrics* | The `$\text{ATT}(g,t)$` framework, DR estimation, IF-based SEs and uniform bands; the `did` package. |
| Sun and Abraham (2021), *J. Econometrics* | The contamination weights on naive TWFE event studies; the interaction-weighted fix; conditions for SA = CS. |
| Borusyak, Jaravel and Spiess (2024), *ReStud* | The imputation estimator and its efficiency under parallel trends; the "forbidden extrapolation" framing. |
| de Chaisemartin and D'Haultfoeuille (2020), *AER* | The switcher estimand `$\text{DID}_M$` and the negative-weights diagnostic for TWFE. |
| Sant'Anna and Zhao (2020), *J. Econometrics* | The doubly-robust DiD moment CS uses: consistent if the outcome *or* the propensity model is right. |
]

**Next module:** Honest DiD. All of the above assume parallel trends holds
exactly. Module 3 asks what survives when it only *almost* holds.

**Drill:** `exercise.R` reproduces CS from scratch (matches `att_gt` to 3e-15),
hand-codes the dCDH switcher, and lines up all four estimators.

---

# Backup Slides

---

# Backup: The Doubly-Robust Estimator

.small[
The unconditional `$\text{ATT}(g,t)$` is a 2x2 of subsample means. With
covariates `$X$` you want robustness to functional-form error. CS uses the
Sant'Anna-Zhao (2020) doubly-robust moment: an outcome-regression model
`$m(X) = \mathbb{E}[\Delta Y \mid X, C]$` for the control trend and a propensity
model `$p(X) = \Pr(G = g \mid X)$` for group membership,

`$$\widehat{\text{ATT}}^{dr}(g,t) = \mathbb{E}\!\left[ \left(\frac{\mathbb{1}\{G=g\}}{\mathbb{E}[\mathbb{1}\{G=g\}]} - \frac{p(X)(1-\mathbb{1}\{G=g\})\,/\,(1-p(X))}{\mathbb{E}\!\left[p(X)(1-\mathbb{1}\{G=g\})\,/\,(1-p(X))\right]}\right) \big(\Delta Y - m(X)\big)\right].$$`

- Consistent if **either** `$m$` or `$p$` is right, not necessarily both: the
  double robustness.
- With no covariates, `$m \equiv$` const and `$p \equiv$` const, and the moment
  collapses to the plain 2x2 difference of subsample means. That identity is
  why `est_method = "dr"` with no `xformla` matches the hand-coded CS to
  machine precision.
- `est_method` alternatives: `"reg"` (outcome only), `"ipw"` (propensity
  only). `"dr"` is the default and the safe choice.
]

---

# Backup: Influence Functions in One Paragraph

.small[
An asymptotically linear estimator can be written
`$\hat\theta - \theta \approx \frac{1}{n}\sum_i \psi(W_i)$`, where `$\psi$` is the
**influence function**: unit `$i$`'s marginal contribution to the estimator. Its
variance is `$\text{Var}(\hat\theta) \approx \frac{1}{n}\mathbb{E}[\psi^2]$`, so
once you have `$\psi_i$` for every observation you have standard errors, and
they correctly account for the estimated nuisance models `$m$` and `$p$` because
`$\psi$` is derived *after* their first-order effect is netted out (Neyman
orthogonality).

The payoff specific to CS: the `$\psi$` for **all** `$(g,t)$` are jointly
available, so you can draw multiplier-bootstrap weights `$\{V_i\}$` (mean 0,
variance 1) and form `$\frac{1}{n}\sum_i V_i \psi_i(g,t)$` simultaneously across
`$(g,t)$`. The sup over `$(g,t)$` of these bootstrap draws gives the critical value
for a band that covers the entire path with the stated probability. That is
why the uniform critical value (2.8 here)
exceeds the pointwise 1.96, and why an honest event-study plot uses it.
]

---

# Backup: The Offset Is a Draw, Not Bias

.small[
The CS estimates on the aggregation table sit above their true targets. Is
that bias? Run the identical CS pipeline on all three effect scenarios and look
at the gap (estimate minus true target):

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> scenario </th>
   <th style="text-align:right;"> simple </th>
   <th style="text-align:right;"> group </th>
   <th style="text-align:right;"> dynamic </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> constant </td>
   <td style="text-align:right;"> 0.254 </td>
   <td style="text-align:right;"> 0.202 </td>
   <td style="text-align:right;"> 0.298 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> dynamic </td>
   <td style="text-align:right;"> 0.254 </td>
   <td style="text-align:right;"> 0.202 </td>
   <td style="text-align:right;"> 0.298 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> heterogeneous </td>
   <td style="text-align:right;"> 0.254 </td>
   <td style="text-align:right;"> 0.202 </td>
   <td style="text-align:right;"> 0.298 </td>
  </tr>
</tbody>
</table>

The gap is the *same* in every scenario (about +0.25 simple, +0.20 group, +0.30
dynamic) because the never-treated group's noise draw is identical across
scenarios (same seed, same 6 cities), and it shifts every `$\text{ATT}(g,t)$`
through the shared base-period term `$\bar Y_{C,g-1}$`. A true bias would scale
with the effect size; a shared draw does not. The fix is a bigger clean control
group (not-yet-treated) or imputation, which is why BJS nails the target.
]

---

# Backup: Where the Contamination Comes From

.small[
Run the naive event-study regression with unit and time FE. By Frisch-Waugh,
the coefficient `$\mu_e$` is a comparison of the cells at relative time `$e$`
against the omitted category, here `$e = -1$` plus the never-treated. When cohorts have
**different** effect paths, the time fixed effect `$\lambda_t$` is itself
contaminated: at calendar time `$t$`, `$\lambda_t$` absorbs a blend of the treated
cohorts' effects present at `$t$`. A unit sitting at event time `$e$` at that same
`$t$` is differenced against this contaminated `$\lambda_t$`, so its coefficient
inherits other cohorts' effects at *their* event times.

Formally, Sun-Abraham show

`$$\mu_e = \sum_{g} \omega_{g,e}\,\delta_{g,e}
\;+\; \sum_{e' \neq e} \sum_{g} \omega_{g,e,e'}\,\delta_{g,e'},$$`

a clean own-event-time piece plus a **cross-event-time** contamination piece.
The `$\omega_{g,e,e'}$` can be negative and do not vanish for `$e < 0$`, so a lead
coefficient loads on other cohorts' lags. Under effect homogeneity all
`$\delta_{g,e} = \delta_e$` and the contamination cancels; under heterogeneity it
does not. The saturated IW estimator sets each `$\delta_{g,e}$` free and averages
with cohort-share weights only, killing the second term.
]

---

# Backup: The Same Comparison, With Noise

.small[
The main event-study slide switched the noise off to isolate contamination.
Here is the honest panel at `$\sigma = 0.3$`, the one you actually estimate:
]

.small[
SA and CS still coincide at every `$e$`. All three lines now wander in the
pre-period because of sampling noise, so a raw pre-trend test cannot separate
contamination from noise: that is exactly why the noiseless panel is the honest
demonstration, and why you report a *band* and stress-test with Honest DiD
rather than eyeballing leads.
]

---

# Backup: Imputation Efficiency, Precisely

.small[
Write the untreated potential outcome as `$Y_{it}(0) = \alpha_i + \lambda_t + \varepsilon_{it}$`. The imputation estimator:

1. **Fit** `$(\hat\alpha, \hat\lambda)$` by OLS on the untreated cells
   `$\mathcal{U} = \{(i,t): t < g_i\}$` (all pre-periods and never-treated).
2. **Impute** `$\hat Y_{it}(0) = \hat\alpha_i + \hat\lambda_t$` on the treated
   cells `$\mathcal{T}$`.
3. **Average** `$\hat\tau_{it} = Y_{it} - \hat Y_{it}(0)$` over `$\mathcal{T}$`
   with target weights.

Why efficient: under parallel trends and homoskedastic `$\varepsilon$`, BJS show
this is the unique linear unbiased estimator that uses *all* untreated cells to
estimate the FE, and it attains the semiparametric variance bound. CS's 2x2
throws away information by anchoring on one base period and one control group;
imputation uses every untreated observation to pin `$\hat\lambda_t$` and
`$\hat\alpha_i$`. On this panel that is a 7.3x
SE reduction (0.033 vs 0.245).

The efficiency comes from trusting the model: the same all-cells fit that
lowers variance also propagates any parallel-trends violation into every
imputed `$\hat Y(0)$`. BJS pair the estimator with a pre-trend test and
recommend the robust (DR or Honest) route when it fails.
]

---

# Backup: dCDH and Negative Weights

.small[
The switcher estimator matters most for the *diagnostic* it comes with. dCDH
(2020) show the TWFE coefficient is `$\sum_{g,t} w_{g,t}\,\text{ATT}(g,t)$` with
weights `$w_{g,t}$` that sum to one but **can be negative**. Negative weights are
what let TWFE report a negative number when every `$\text{ATT}(g,t) > 0$`
(Module 1's sign-reversal panel).

The `DIDmultiplegt` package reports, for any TWFE spec, the number and mass of
negative weights and the "minimal standard deviation of the ATTs under which
TWFE and the true ATT have opposite signs". A tiny such number is a red flag.

The full `$\text{DID}_M$` handles treatment that switches **on and off**, which
is where CS/SA/BJS (built for absorbing, staggered adoption) do not directly
apply. In our no-exit rollout only the instantaneous piece is identified from
switch events; the hand-coded version recovers it:

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> switch time </th>
   <th style="text-align:right;"> n switchers </th>
   <th style="text-align:right;"> DID_M </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 15 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 0.732 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 25 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 0.305 </td>
  </tr>
  <tr>
   <td style="text-align:right;"> 35 </td>
   <td style="text-align:right;"> 8 </td>
   <td style="text-align:right;"> 0.239 </td>
  </tr>
</tbody>
</table>

Per-switch estimates are noisy (small stable-control groups at late switch
times), but the switcher-weighted overall (0.425) recovers
the true instantaneous effect of 0.400.
]

---

# Backup: The Shared 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))   # PT by construction
}
```

Identical to Module 1, so the numbers are comparable across decks. City FE and
a common weekly trend make parallel trends hold **by construction**: every gap
between an estimate and its true target on these slides is either the
estimator's design (TWFE) or finite-sample noise in a small control group (CS),
never a real parallel-trends violation. The `heterogeneous` scenario scales the
early cohort up 1.8x and the late cohort down to 0.2x, which is what makes the
aggregations disagree and TWFE collapse to 0.42.
]