Module 3: Honest DiD

.title[
# Module 3: Honest DiD
]
.subtitle[
## Sensitivity Bounds for Parallel Trends: Rambachan-Roth
]

---

# 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>2</td><td><a href="../module-02/slides.html">Heterogeneity-Robust DiD: CS, SA, BJS, dCDH</a></td><td>done</td></tr>
<tr><td><b>3</b></td><td><b>Honest DiD: Sensitivity Bounds for Parallel Trends</b> <i>(you are here)</i></td><td>current</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

Modules 1 and 2 assumed parallel trends and fixed the *staggered-timing* bias.
This module stress-tests parallel trends itself. The tour said pre-trend tests
are weak and Rambachan-Roth bound the damage. Here is the machinery.

**You will be able to:**

1. Say precisely why a passing pre-trend test is weak evidence: low power and
   pre-test bias, both from Roth (2022).
2. Write the Rambachan-Roth (2023) restriction sets `$\Delta^{SD}(M)$` and
   `$\Delta^{RM}(\bar M)$` and say what economic story each encodes.
3. Report a **breakdown value**: how large a violation it takes to overturn a
   result.
4. Run the `HonestDiD` workflow end to end and read the sensitivity plot.

**Running application:** the single-cohort event study carved out of the
Module 1-2 rollout: cohort `$g = 25$` versus six never-treated cities. Outcome:
log completed trips (read: engagement or revenue per customer in a retail
setting). The rollout was never randomized, so parallel trends is an
assumption you defend, not a design you can lean on; this module is how
you quantify exactly how much you are leaning on it.

---

# A Passing Pre-Trend Test Is Weak Evidence

.pull-left[
The event-study "pre-trend test" is a joint Wald test that the lead
coefficients are zero. Two failures, both Roth (2022):

- **Low power.** A near-linear violation is the exact shape the test cannot
  see. Below, a slope that biases the estimate is caught well under 80% of
  the time.
- **Pre-test bias.** Dropping analyses that fail the test does not clean up
  the survivors. It makes the bias **worse** (next).
]

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

.highlight-box[
Same design as our application. A slope `$\delta = 0.10$` biases the naive
first-post estimate by that much yet trips the test only about
55%
of the time.
]

<a href="#pretrend-power-detail" class="nav-btn">power derivation</a>
<a href="#selective-inference-detail" class="nav-btn-br">pre-test bias</a>

---

# The Rambachan-Roth Reframe

Stack the event-study coefficients, `$\hat\beta \sim \mathcal{N}(\beta, \Sigma)$`,
and split each into effect and violation:

`$$\beta_{post} = \underbrace{\tau_{post}}_{\text{causal effect}} + \underbrace{\delta_{post}}_{\text{PT violation}}, \qquad \beta_{pre} = \delta_{pre}.$$`

Exact PT is `$\delta = 0$`. **Drop it.** Assume only `$\delta \in \Delta$` for a
set `$\Delta$` you choose.

The pre-periods estimate `$\delta_{pre}$` directly, so `$\Delta$` links the
*observed* pre-violation to the *unobserved* post-violation. The target
`$\theta = \ell' \tau_{post}$` is then **partially identified**:

`$$\mathcal{S}(\beta, \Delta) = \left\{ \ell'(\beta_{post} - \delta_{post}) : \delta \in \Delta,\ \delta_{pre} = \beta_{pre} \right\}.$$`

.blue-box[
Honesty is paid in **width**. As `$\Delta$` grows the identified set widens and
the robust CI grows with it. The point estimate does not silently move; the
uncertainty does.
]

<a href="#partial-id-detail" class="nav-btn">partial identification</a>

---

# The Restriction Menu

Choosing `$\Delta$` *is* the method. Each choice is an economic claim about the
confounder you fear.

.small[
| `$\Delta$` | Restriction | Story |
|---|---|---|
| `$\Delta^{SD}(M)$` | second differences of `$\delta$` bounded by `$M$` | trend does not *accelerate* by more than `$M$`/period |
| `$\Delta^{RM}(\bar M)$` | post jump `$\leq \bar M \times$` worst pre jump | post violation no worse than pre, scaled |
| `$\Delta^{SDPB}, \Delta^{RMB}$` | `$+$` sign of `$\delta_{post}$` | you know the direction of the bias |
| `$\Delta^{SDM}, \Delta^{SDI}$` | `$+$` monotone `$\delta$` | the trend does not reverse |
]

.highlight-box[
Impose only what institutional knowledge defends. A sign restriction you
cannot justify buys a narrower interval you should not trust. Sign and
monotonicity *combine* with `$\Delta^{SD}$` or `$\Delta^{RM}$`; each added
restriction shrinks the identified set.
]

---

# Smoothness: `$\Delta^{SD}(M)$`

Bound the discrete second difference of the violation:

`$$\Delta^{SD}(M) = \Big\{ \delta : \big| (\delta_{s+1} - \delta_s) - (\delta_s - \delta_{s-1}) \big| \leq M \ \ \forall s \Big\}.$$`

- `$M = 0$`: violation is exactly **linear**, extrapolated from the pre-period
  slope.
- `$M > 0$`: allow curvature up to `$M$` per period.

.blue-box[
Every **linear** trend has zero second difference, so it sits in
`$\Delta^{SD}(M)$` for *all* `$M \geq 0$`. Under `$\Delta^{SD}$` the estimator
extrapolates the pre-period line into the post-period and nets it out. A purely
linear confound is something `$\Delta^{SD}$` **corrects for**, not something it
flags.
]

Right when your worry is a smooth secular trend (adoption momentum, a slow
demand shift) contaminating the comparison.

<a href="#delta-sd-derivation" class="nav-btn">geometry + why linear is free</a>

---

# Relative Magnitudes: `$\Delta^{RM}(\bar M)$`

Bound the largest post jump by `$\bar M$` times the largest pre jump:

`$$\Delta^{RM}(\bar M) = \Big\{ \delta : |\delta_{s+1} - \delta_s| \leq \bar M \cdot \max_{s' < 0} |\delta_{s'+1} - \delta_{s'}| \ \ \forall s \geq 0 \Big\}.$$`

The story: whatever moved the groups apart before treatment cannot suddenly
move them much faster after. `$\bar M = 1$` means the post violation is no bigger,
period for period, than the worst wiggle already visible pre-treatment.

.highlight-box[
Unlike `$\Delta^{SD}$`, this keys on the **magnitude** of the estimated
pre-violation, so it *does* react to a linear confound: a steeper pre-trend
raises the benchmark and widens the bounds. The default when you will not assume
the trend is smooth, only that it does not change character at the treatment
date.
]

<a href="#delta-rm-derivation" class="nav-btn">the RM set, formally</a>

---

# The Breakdown Value

Do not pick one `$M$`. Trace the robust CI as `$M$` grows and report where the
conclusion flips:

`$$M^{\ast} = \sup \Big\{ M : 0 \notin \mathcal{C}_{1-\alpha}\big(\Delta^{SD}(M)\big) \Big\},$$`

the largest smoothness allowance under which the robust CI still excludes zero
(and `$\bar M^{\ast}$` for `$\Delta^{RM}$`).

.blue-box[
It converts "is the effect significant?" into "**how large a PT violation would
it take to make it insignificant?**" You then judge whether a violation that
large is plausible, using the observed pre-period wiggle as the yardstick.
`$\bar M^{\ast} = 1.5$` reads directly: the post violation would have to be 1.5x
the worst pre one.
]

In a decision memo for a non-technical owner, the breakdown value collapses
to one sentence: "the effect survives unless post-launch confounding is
1.5x the worst pre-period wiggle."

<a href="#breakdown-derivation" class="nav-btn">reading it off the CI path</a>

---

# How the Robust CI Is Built

.small[
| Method | Idea | Use for |
|---|---|---|
| **FLCI** | shortest fixed-length interval, min-max over `$\Delta$` | convex, centrosymmetric `$\Delta$` (e.g. `$\Delta^{SD}$`) |
| **Conditional / C-LF** | invert an ARP moment-inequality test over `$\mathcal{S}$` | non-centrosymmetric `$\Delta$` (e.g. `$\Delta^{RM}$`) |
]

The conditional least-favorable hybrid (**C-LF**) of Andrews-Roth-Pakes is the
recommended default for `$\Delta^{RM}$`, where FLCI does not apply.

.highlight-box[
On this machine the FLCI solver is unavailable, so every run here uses
**C-LF**, which is valid for both sets. For `$\Delta^{SD}$` the FLCI is usually a
touch shorter; the breakdown-value logic is identical.
]

---

# Application: One Clean Cohort

From the Module 1-2 rollout, carve out cohort `$g = 25$` (8 cities) versus the 6
never-treated cities, window `$-5$` to `$+5$`, clustered by city.

```r
d <- make_panel("constant") |>
  filter(g == 25 | !is.finite(g)) |>                  # one cohort + controls
  mutate(rel_time = if_else(is.finite(g), t - 25L, -1L)) |> # never-tr -> base
  filter(t >= 20, t <= 30)                            # rel_time in [-5, 5]
feols(y ~ i(rel_time, ref = -1) | city + t, data = d, cluster = ~city)
```

.highlight-box[
A clustered `$\Sigma$` has rank at most (clusters `$- 1$`). With 14 clusters, a
`$-10..+10$` window (20 coefficients) gives a rank-deficient `$\Sigma$` and the
bounds explode. The `$-5..+5$` window (10 coefficients) stays under the ceiling.
Always check the coefficient count against the cluster count.
]

<a href="#application-dgp" class="nav-btn">event-study construction</a>

---

# The Same Problem at an Online Retailer

- Metros were upgraded earliest where demand growth was strongest, making the rollout order endogenous: parallel trends is suspect on its face.
- High-growth metros would have seen rising orders even without next-day delivery; the pre-period trend in early-adopting metros diverges from the never-upgraded group before the rollout begins.
- Rambachan-Roth: bound the post-period violation by a multiple `$\bar{M}$` of the worst pre-period divergence; the robust confidence set for the delivery effect widens with `$\bar{M}$` rather than collapsing to a false point estimate.
- The breakdown value gives the minimum pre-trend-relative violation that would make the delivery effect statistically indistinguishable from zero.
- Reporting the robust confidence set alongside the TWFE point estimate turns an endogenously ordered rollout into a credible study, rather than a claim that parallel trends holds by assumption.

---

# The Event Study

Parallel trends holds by construction; the pre-test passes, with
`$p = 0.16$`. But "passes" is exactly what Roth warns us to distrust.

---

# First, Validate the Handoff

`HonestDiD` wants `betahat` in order (earliest pre `$\dots -2$`, then `$0 \dots K$`)
and its `sigma`. `fixest`'s `i()` already returns that order. **Confirm
`constructOriginalCS` reproduces the `feols` CI before trusting anything.**

```r
ocs1 <- constructOriginalCS(es$beta, es$sigma,        # numPre = 4, numPost = 6
                            es$numPre, es$numPost, l_vec = basisVector(1, es$numPost))
c(honestdid_lb = ocs1$lb, honestdid_ub = ocs1$ub,
  feols_beta   = es$beta[which(es$evt == 0)])
```

```
## honestdid_lb honestdid_ub   feols_beta 
##    0.4571607    1.3187324    0.8879465
```

The `HonestDiD` "original" CI is [0.457, 1.319],
matching `$\hat\beta_0 \pm 1.96\,\mathrm{SE}$` for the first post-period exactly.
Ordering is right; now the sensitivity results mean something.

---

# The Workflow: Event Study In, Sensitivity Out

```r
# smoothness: second differences bounded by each M
sd <- createSensitivityResults(
  betahat = es$beta, sigma = es$sigma,
  numPrePeriods = 4, numPostPeriods = 6,
  Mvec = seq(0.05, 0.6, 0.05),
  l_vec = basisVector(1, 6),          # target the FIRST post-period
  method = "C-LF")                    # FLCI solver unavailable here

# relative magnitudes: post jump <= Mbar * worst pre jump
rm <- createSensitivityResults_relativeMagnitudes(
  betahat = es$beta, sigma = es$sigma,
  numPrePeriods = 4, numPostPeriods = 6,
  Mbarvec = seq(0.25, 2, 0.25), l_vec = basisVector(1, 6))
```

Two calls, one per restriction. Each returns the robust CI at every `$M$` (or
`$\bar M$`) in the grid. Keep the RM grid short: it is the slower of the two.

---

# Smoothness Sensitivity, Clean Case

The robust CI first touches zero near `$M = 0.45$`:
that is the smoothness breakdown value `$M^{\ast}$`.

---

# Relative-Magnitudes Sensitivity, Clean Case

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

.small[
| Restriction | Breakdown |
|---|---|
| `$\Delta^{SD}$` | `$M^{\ast} = 0.45$` |
| `$\Delta^{RM}$` | `$\bar M^{\ast} = 1.50$` |
]

Here the relative-magnitudes breakdown is 1.5, so
the post violation would have to be 1.5x the worst
pre-period wiggle to kill significance. Both are large relative to anything the
pre-period shows.
]

.blue-box[
**Report a number, not a verdict.** "Significant, and it survives a
parallel-trends violation 1.5x the pre-period one"
beats "the pre-trends looked flat."
]

---

# The Target `$\ell$` Matters

Switch `$\ell$` from the first post-period to the **average** of all six. The same
data, a different estimand:

.small[
| Target `$\ell$` | Original CI | `$M^{\ast}$` | `$\bar M^{\ast}$` |
|---|---|---|---|
| first post-period | [0.46, 1.32] | 0.45 | 1.50 |
| average of post | [0.83, 1.49] | 0.05 | 0.50 |
]

.highlight-box[
The average is far more fragile. Under `$\Delta^{SD}$` the worst-case bias at
post-period `$s$` grows like `$M \cdot s^2$`, so an average loading on late horizons
is exposed to compounding extrapolation. The first post-period is the most
robust target; long-run averages the least. Pick `$\ell$` to match the estimand
you report, and expect later horizons to be more fragile.
]

<a href="#l-vec-detail" class="nav-btn">the compounding math</a>

---

# Confounded Case: the Restriction Must Match the Threat

Inject a differential linear trend of slope `$0.10$` into the treated cohort. The
pre-test now rejects sharply, with `$p = 0.0001$`. The breakdown
values split:

.small[
| Restriction | Clean | Confounded |
|---|---|---|
| `$\Delta^{SD}$` | `$M^{\ast} = 0.45$` | `$M^{\ast} = 0.45$` |
| `$\Delta^{RM}$` | `$\bar M^{\ast} = 1.50$` | `$\bar M^{\ast} = 1.00$` |
]

.highlight-box[
`$\Delta^{RM}$` drops: the steeper pre-violation inflates the benchmark. `$\Delta^{SD}$`
**does not move**: a linear trend has zero second difference, so smoothness
extrapolates and nets it out. If you fear a smooth confound, `$\Delta^{SD}$`
*assumes it continues and corrects it* (strong). If you will not assume that,
`$\Delta^{RM}$` is the honest choice.
]

<a href="#sd-vs-rm-detail" class="nav-btn">why SD is blind to a line</a>

---

# One Rejection Is Not a Reliable Test

In that confounded sample the pre-test rejected. So it "worked." But that is
**one draw**.

- Whether the test *reliably* catches a trend of slope `$0.10$` is a
  repeated-sampling property.
- The power curve on slide 3 says it is caught only about
  55%
  of the time.
- And conditioning on the ~45%
  that pass makes the surviving bias **worse**, because leads and lags are
  positively correlated, about `$+0.49$` here.

.blue-box[
The gap between "this sample rejected" and "the test reliably rejects" is the
entire Roth (2022) argument, and the reason the deliverable is a breakdown value,
not a pre-test `$p$`. `exercise.R` reproduces both facts analytically.
]

<a href="#selective-inference-detail" class="nav-btn">conditioning is worse</a>

---

# Practitioner Checklist

.small[
| Step | What you do | Why |
|---|---|---|
| 1. Dimension | coefficient count `$<$` cluster count | else `$\Sigma$` is rank-deficient, bounds explode |
| 2. Validate | `constructOriginalCS` `$=$` `feols` CI | catches a mis-ordered `betahat` |
| 3. Choose `$\Delta$` | `$\Delta^{SD}$` for smooth, `$\Delta^{RM}$` for "no worse than pre" | the restriction is an economic claim |
| 4. Breakdown | report `$M^{\ast}$` / `$\bar M^{\ast}$`, calibrate vs pre-period | one honest number |
| 5. Plot | show the sensitivity path, not a pass/fail | conveys the whole robustness curve |
]

At platform scale (staggered launches across marketplaces, fulfillment centers,
categories), a clean never-treated control and long pre-periods are rare, so
"how robust is this to the parallel-trends assumption?" is asked of every
DiD readout. This is the answer.

---

# Interview Questions

.small[
| Question | Core of a strong answer |
|---|---|
| "The pre-trends look flat. Are we safe?" | Flat pre-trends are weak evidence: the test has low power against near-linear violations and pre-testing biases the survivors. Report a breakdown value, not a pass/fail. |
| "What is a breakdown value and how do I read it?" | The largest PT violation (in `$M$` or `$\bar M$`) under which the robust CI still excludes zero. `$\bar M^{\ast} = 1.5$`: the post violation would need to be 1.5x the worst pre one. |
| "Smoothness or relative magnitudes?" | `$\Delta^{SD}$` if you fear a smooth trend (it extrapolates and corrects a linear one); `$\Delta^{RM}$` if you only assume the post violation is no worse than the pre. Match the restriction to the confounder. |
| "A linear confound with a clean breakdown value under `$\Delta^{SD}$`: trust it?" | Only if a *linear* trend is truly your worry: `$\Delta^{SD}$` nets linear trends out, so its `$M^{\ast}$` is blind to them. Check `$\Delta^{RM}$`, which reacts. |
| "How does this scale to our launches?" | Few clusters cap the event-study width (rank of `$\Sigma$`); validate the original CI; report breakdown values per horizon since later horizons are more fragile. |
| "So is the result robust or not? I need a yes or no." | Translate the breakdown value into one sentence: the effect survives unless post-launch confounding is 1.5x the worst pre-period wiggle. Then name the candidate shocks and say whether one that large is plausible. |
]

---

# Going Deeper

.small[
| Paper | What it adds |
|---|---|
| Rambachan and Roth (2023), *ReStud* | The framework on these slides: the sets `$\Delta$`, partial identification, robust CIs, breakdown values. |
| Roth (2022), *AER: Insights* | Pre-trend tests have low power; conditioning on passing distorts inference. The exercise reproduces both. |
| Andrews, Roth and Pakes (2023), *ReStud* | The conditional moment-inequality inference (ARP / C-LF) behind the robust CIs. |
| Roth, Sant'Anna, Bilinski and Poe (2023), *J. Econometrics* | Where honest DiD sits in the modern DiD toolbox. |
| Manski (2003) | The partial-identification tradition the whole approach descends from. |
]

**Next module:** synthetic control, a different answer to "what is the
counterfactual trend?" that reweights donors instead of assuming parallelism.

**Drill:** `exercise.R` reproduces the Roth power simulation analytically and
computes breakdown values, clean vs confounded.

---

# Backup Slides

---

# Backup: Roth's Power Calculation, Analytically

.small[
No panel re-simulation needed. Fit the event study **once** to get `$\Sigma$`, then
draw `$\hat\beta \sim \mathcal{N}(\beta_{true}, \Sigma)$`. Impose a linear
violation of slope `$\delta$` and no true effect:

`$$\beta_{true, s} = \delta \cdot (s + 1) \quad (\text{relative to base period } -1).$$`

For each draw, the pre-trend Wald statistic is
`$\hat\beta_{pre}' \Sigma_{pre}^{-1} \hat\beta_{pre}$`; power is the share exceeding
`$\chi^2_{0.95, k_{pre}}$`.

```r
roth_sim <- function(delta, R = 3000) {
  bt <- delta * (evt + 1)                         # linear trend, tau = 0
  dr <- MASS::mvrnorm(R, mu = bt, Sigma = Sigma)
  wald <- apply(dr[, pre_idx], 1, function(b) t(b) %*% solve(Sigma_pre) %*% b)
  mean(wald > qchisq(0.95, df = numPre))          # power
}
```

Because `$\hat\beta_{pre}$` is Gaussian, this is exact, not Monte Carlo error on a
DGP. The worst-case linear slope is the one the test is *least* able to see, and
that is precisely the slope that biases the post estimate most. Low power is
structural, not a small-sample accident.
]

---

# Backup: Why Conditioning Makes Bias Worse

.small[
The naive first-post estimate is unbiased *for the biased estimand*
`$\tau + \delta$`: unconditionally `$\mathbb{E}[\hat\beta_0] = \delta$` (here
`$\tau = 0$`). A "pre-test estimator" keeps only draws that pass, i.e. draws whose
`$\hat\beta_{pre}$` landed near zero.

The violation pushes `$\beta_{pre}$` *away* from zero, so a draw passes when noise
pushes it *back*. Leads and lags share the base period `$-1$`, so
`$\mathrm{Cov}(\hat\beta_{pre}, \hat\beta_0) > 0$`, about
`$+0.49$` here. The same noise that shrank the leads
inflated the lag:

`$$\big|\mathbb{E}[\hat\beta_0 \mid \text{pass}]\big| \; > \; \big|\mathbb{E}[\hat\beta_0]\big|.$$`

| `$\delta$` | naive bias | bias `$\mid$` passed |
|---|---|---|
| 0.06 | 0.052 | 0.096 |
| 0.10 | 0.093 | 0.206 |

At `$\delta = 0.10$` pre-testing roughly **doubles** the bias. Screening on
pre-trends does not rescue you; it selects the samples where the trend hid.
]

---

# Backup: The Identified Set, Concretely

.small[
Only the sum `$\tau_{post} + \delta_{post}$` is observed. Fix the pre-period at its
estimate, `$\delta_{pre} = \beta_{pre}$`, and sweep `$\delta_{post}$` over everything
`$\Delta$` permits given that pre-period. The set of target values consistent with
the data is

`$$\mathcal{S}(\beta, \Delta) = \Big\{ \ell'\big(\beta_{post} - \delta_{post}\big) : (\beta_{pre}, \delta_{post}) \in \Delta \Big\}.$$`

- If `$\Delta = \{0\}$` (exact PT), `$\mathcal{S}$` is the point `$\ell'\beta_{post}$`:
  the usual event-study estimate.
- If `$\Delta = \mathbb{R}^{k}$` (anything goes), `$\mathcal{S} = \mathbb{R}$`:
  nothing is identified without a control-trend assumption.
- The useful sets sit in between: big enough to be credible, small enough to say
  something.

Inference then covers the *set*, not a point. Two flavors: cover the whole
identified set, or cover the true parameter (the ARP conditional approach, less
conservative). `HonestDiD` does the latter.
]

---

# Backup: `$\Delta^{SD}$` Geometry and the Free Linear Trend

.small[
Write the violation path `$\delta_{-k}, \dots, \delta_{-1}=0, \delta_0, \dots$`.
The second difference `$\Delta^2 \delta_s = \delta_{s+1} - 2\delta_s + \delta_{s-1}$`
is the discrete curvature. `$\Delta^{SD}(M)$` caps `$|\Delta^2 \delta_s| \leq M$`
everywhere.

Decompose any path into a **linear** part (fixed by the two base-adjacent
points) plus curvature. `$\Delta^2$` annihilates the linear part, so:

- The linear component is *unrestricted* by `$\Delta^{SD}(M)$` for any `$M$`. The
  identified set is centered on the linear extrapolation of the pre-trend.
- `$M$` controls only how far the post path may bow away from that line. The
  worst-case deviation at post-period `$s$` grows like `$M \cdot s^2$` (curvature
  integrated twice), which is why late horizons blow up first.

Consequence: shifting the whole sample by a linear confound moves the estimate
and the extrapolation **together**, leaving `$M^{\ast}$` unchanged (slide
[confounded case](#confounded-main)). `$\Delta^{SD}$` answers "how much curvature
can I tolerate?", never "is there a line?".
]

---

# Backup: `$\Delta^{RM}$`, Formally

.small[
Let the first difference be `$\Delta \delta_s = \delta_{s+1} - \delta_s$` (the
period-to-period change in the violation). Define the worst pre-period change

`$$D_{pre} = \max_{s' < 0} |\Delta \delta_{s'}|,$$`

which the pre-period coefficients estimate. `$\Delta^{RM}(\bar M)$` requires every
post-period change to obey

`$$|\Delta \delta_s| \leq \bar M \cdot D_{pre}, \qquad s \geq 0.$$`

- The set is a scaled cone anchored to the *observed* pre-period wiggle, not an
  absolute magnitude, hence "relative."
- It is **not centrosymmetric**, so FLCI does not apply and inference uses the
  ARP conditional / C-LF construction.
- A steeper pre-trend raises `$D_{pre}$`, enlarges the set, and lowers
  `$\bar M^{\ast}$`: the restriction reacts to exactly the linear confound that
  `$\Delta^{SD}$` ignores.

Reach for `$\Delta^{RM}$` when you are unwilling to assume smoothness but will
assume the post-period trend is no wilder than the pre-period one.
]

---

# Backup: Reading the Breakdown Value Off the CI Path

.small[
For each `$M$`, `HonestDiD` returns a robust CI `$[\text{lb}(M), \text{ub}(M)]$`.
Both endpoints move monotonically outward in `$M$` (a bigger `$\Delta$` can only
enlarge the identified set). Define the breakdown value as the first `$M$` at
which the CI covers zero:

`$$M^{\ast} = \min\{ M : \text{lb}(M) \leq 0 \leq \text{ub}(M) \}.$$`

Practically, run a grid and find where `$\text{lb}(M)$` crosses zero. In the clean
case, `$\text{lb}$` falls from 0.29 at `$M = 0.05$` through
zero near `$M = 0.45$`:

```
## # A tibble: 10 × 4
##        M     lb    ub excludes_0
##    <dbl>  <dbl> <dbl> <lgl>     
##  1  0.05  0.289  1.96 TRUE      
##  2  0.1   0.248  2.00 TRUE      
##  3  0.15  0.214  2.03 TRUE      
##  4  0.2   0.17   2.08 TRUE      
##  5  0.25  0.135  2.12 TRUE      
##  6  0.3   0.089  2.16 TRUE      
##  7  0.35  0.052  2.2  TRUE      
##  8  0.4   0.005  2.24 TRUE      
##  9  0.45 -0.034  2.28 FALSE     
## 10  0.5  -0.083  2.33 FALSE
```

Report `$M^{\ast}$` with its yardstick: "the effect survives curvature up to
`$M^{\ast}$`, versus a pre-period second difference of at most (observed value)."
A breakdown value without a scale to compare it to is not yet an argument.
]

---

# Backup: FLCI vs the ARP Conditional Approach

.small[
**FLCI** (fixed-length CI). Restrict to affine estimators `$a + v'\hat\beta$` and
choose `$v$` to minimize the CI length subject to worst-case coverage over
`$\delta \in \Delta$`. This is a tractable convex program **only when `$\Delta$` is
convex and centrosymmetric**, i.e. `$\delta \in \Delta \Rightarrow -\delta \in \Delta$`,
which `$\Delta^{SD}(M)$` satisfies but `$\Delta^{RM}(\bar M)$` does not. FLCI gives the
shortest honest interval when it applies.

**ARP conditional (Andrews-Roth-Pakes).** Cast `$\delta \in \Delta$` as linear
moment inequalities and invert a test of `$H_0: \theta = \theta_0$` over a grid of
`$\theta_0$`. The conditional test corrects for the fact that which inequalities
bind is data-dependent. The **least-favorable hybrid (C-LF)** blends a
least-favorable first stage with the conditional test for power. It is valid for
*any* polyhedral `$\Delta$`, so it covers `$\Delta^{RM}$` and the combined sets.

Rule of thumb: `$\Delta^{SD}$` alone -> FLCI (shortest); anything with relative
magnitudes, signs, or monotonicity -> C-LF. Here FLCI is unavailable, so C-LF is
used throughout, at a small length cost for `$\Delta^{SD}$`.
]

---

# Backup: Building the Event Study

```r
event_fit <- function(scenario = "constant", confound = 0) {
  d <- make_panel(scenario) |>
    filter(g == 25 | !is.finite(g)) |>                     # one cohort + never-tr
    mutate(rel_time = if_else(is.finite(g), as.integer(t - 25), -1L)) |>
    filter(t >= 20, t <= 30)                               # rel_time in [-5, 5]
  if (confound != 0)                                       # optional PT violation
    d <- d |> mutate(y = y + if_else(g == 25, confound * (t - 25), 0))
  feols(y ~ i(rel_time, ref = -1) | city + t, data = d, cluster = ~city)
}
tidy_es <- function(fit) {                                 # HonestDiD-ready pieces
  ev <- as.integer(sub("rel_time::", "", names(coef(fit)))); ord <- order(ev)
  list(beta = unname(coef(fit)[ord]), sigma = vcov(fit)[ord, ord],
       evt = ev[ord], numPre = sum(ev < 0), numPost = sum(ev >= 0))
}
```

- Never-treated cities get `rel_time = -1` so they land in the base category and
  serve as controls at every week through the time fixed effects.
- `filter(t >= 20, t <= 30)` matches the calendar window to `$[-5, +5]$` event
  time for the `$g = 25$` cohort, so the time FE are estimated only where the
  treated cohort has an event-time coefficient.
- `order(ev)` guarantees the (pre earliest `$\dots$` post) ordering `HonestDiD`
  expects. This is the step to validate against the `feols` CI.
]

---

# Backup: Why the Average Target Is Fragile

- **First post-period:** `$\ell = (1, 0, 0, 0, 0, 0)'$`. Under `$\Delta^{SD}(M)$`
  the worst-case bias is bounded by `$M \cdot 1^2$`: only one period of
  extrapolation.
- **Average of post:** `$\ell = (\tfrac16, \dots, \tfrac16)'$`. The bias at
  horizon `$s$` is bounded by `$\sim M \cdot s^2$`, so the average is
  `$\frac{M}{6}\sum_{s=0}^{5} s^2 \approx M \cdot 9$`: nine times the exposure.

That is why the same panel gives `$M^{\ast} = 0.45$`
for the first period but `$M^{\ast} = 0.05$` for the
average, and `$\bar M^{\ast}$` falls from 1.50 to
0.50. The average is not "wrong": it answers a
different, later-horizon question that is intrinsically harder to defend against
trend extrapolation.

Practical rule: report the breakdown value **for the horizon you will act on**.
If the decision hinges on the long-run average, its fragility is real
information, not a nuisance.
]

---

# Backup: The Same Confound, Two Verdicts

.small[
The injected confound is `$0.10 \cdot (t - 25)$` added to the treated cohort: a
pure line through the treatment date. In event-study coordinates it shifts every
coefficient by a linear amount.

| | clean | confounded |
|---|---|---|
| pre-test `$p$` | 0.16 | 0.0001 |
| `$M^{\ast}$` under `$\Delta^{SD}$` | 0.45 | 0.45 |
| `$\bar M^{\ast}$` under `$\Delta^{RM}$` | 1.50 | 1.00 |

- **Smoothness does not react.** A line has zero second difference, so it is
  inside `$\Delta^{SD}(0)$`. The estimator extrapolates the pre-period line into
  the post and subtracts it; the estimate and the correction move together and
  `$M^{\ast}$` is untouched. `$\Delta^{SD}$` is *defined* to be blind to a linear
  confound.
- **Relative magnitudes does react.** The confound enlarges the worst pre-period
  jump `$D_{pre}$`, so `$\Delta^{RM}(\bar M)$` permits a proportionally larger post
  violation and the bounds widen: `$\bar M^{\ast}$` falls
  1.50 `$\to$` 1.00.

Neither is "more correct." They encode different beliefs. The takeaway: never
read a breakdown value without stating which `$\Delta$` produced it and whether
that `$\Delta$` can even see the confound you fear.
]