Module 5: Synthetic DiD

.title[
# Module 5: Synthetic DiD
]
.subtitle[
## The Bridge from Synthetic Control to DiD (Arkhangelsky et al. 2021)
]

---

# 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>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><b>5</b></td><td><b>Synthetic DiD and the Bridge from SC to DiD</b> <i>(you are here)</i></td><td>current</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 4 built synthetic control by reweighting *donors*. Module 1
diagnosed DiD as a global parallel-trends bet. Arkhangelsky et al. (2021)
show these are two corners of **one** estimator, and the interior is better
than either.

**You will be able to:**

1. Write the SDID objective as a weighted TWFE regression, and state both
   weight programs plus the regularization `$\zeta$` from memory.
2. Locate DiD, SC, and SDID in the unit-weight `$\times$` time-weight square,
   and say precisely what turns one into another.
3. Explain why the unit intercept is the single change that separates SDID
   from pure SC.
4. Compute all three on Prop 99, read the signature plot, and pick the right
   variance estimator.

**Applications:** California Prop 99 (the canonical comparison), then a
ride-share factor-structure panel (outcome: a rides index, read units sold
or sessions per market in a retail setting) where DiD breaks and SDID
recovers truth. One treated market again: there is no room to randomize a
policy that only ever touched one city.

---

# Two Estimators, One Weakness Each

`$$\hat\tau = \Delta\bar y_{\text{treat}} - \Delta\bar y_{\text{ctrl}}$$`

- Uses *every* control, weighted equally.
- Bets on **global** parallel trends.
- Breaks when the raw control average is on a different path.
]

`$$\hat y^N_{1t} = \sum_i \hat\omega_i\, y_{it}$$`

- Reweights donors to match **pre-period levels**.
- No time reweighting; must fit levels exactly.
- Fragile when no donor mix reaches the treated level.
]

.highlight-box[
SDID keeps SC's unit reweighting, **adds** a time reweighting, and **frees**
the level match with an intercept. The result is a weighted DiD that is
consistent if *either* set of weights works: double robustness in the panel.
]

---

# The SDID Estimator

Block treatment: last `$N_1 = N - N_0$` units treated in the last
`$T_1 = T - T_0$` periods, `$W_{it} = 1$` on treated cells. SDID solves a
**weighted** two-way fixed effects problem:

`$$(\hat\tau, \hat\mu, \hat\alpha, \hat\beta) =
\arg\min_{\tau, \mu, \alpha, \beta}
\sum_{i=1}^{N}\sum_{t=1}^{T}
\left(Y_{it} - \mu - \alpha_i - \beta_t - \tau W_{it}\right)^2
\,\hat\omega_i\, \hat\lambda_t$$`

Three moving parts, in order:

1. **Unit weights `$\hat\omega_i$`** on the `$N_0$` controls: a synthetic-control
   match, but with an intercept and ridge penalty.
2. **Time weights `$\hat\lambda_t$`** on the `$T_0$` pre-periods: which "before"
   periods to compare against.
3. **Plug in**, run weighted TWFE. Given the weights, `$\hat\tau$` is a
   *weighted double difference*.

.blue-box[
Compare the identical objective with `$\hat\omega_i = 1/N$`, `$\hat\lambda_t = 1/T$`:
that is ordinary TWFE, i.e. DiD. SDID is DiD with the weights chosen,
not assumed.
]

<a href="#reconstruct-detail" class="nav-btn">the double-difference identity</a>

---

# The Unit-Weight Program

Choose donor weights so the weighted control **tracks the treated trend**
across the pre-period, up to a level shift `$\omega_0$`:

`$$(\hat\omega_0, \hat\omega) = \arg\min_{\omega_0 \in \mathbb{R},\, \omega \in \Omega} \sum_{t=1}^{T_0} \Big(\omega_0 + \sum_{i=1}^{N_0}\omega_i Y_{it} - \tfrac{1}{N_1}\!\!\sum_{i=N_0+1}^{N}\! Y_{it}\Big)^2 + \zeta^2 T_0 \lVert\omega\rVert_2^2$$`

over the simplex `$\Omega = \{\omega \geq 0, \sum_i \omega_i = 1\}$`.

Two departures from Module 4's SC program:

- **Intercept `$\omega_0$`** (free, off the simplex): donors match *shape*,
  not *level*. Level gaps are absorbed here and by the unit FE `$\alpha_i$`.
- **Ridge `$\zeta^2 T_0 \lVert\omega\rVert_2^2$`**: spreads weight across
  donors, guarantees a unique solution under collinearity.

<a href="#omega-derivation" class="nav-btn">solve it with quadprog</a>
<a href="#zeta-detail" class="nav-btn-br">where `$\zeta$` comes from</a>

---

# The Time-Weight Program

Symmetric: choose pre-period weights so the weighted **pre-periods resemble
the post-period** for the controls:

`$$(\hat\lambda_0, \hat\lambda) = \arg\min_{\lambda_0 \in \mathbb{R},\, \lambda \in \Lambda} \sum_{i=1}^{N_0} \Big(\lambda_0 + \sum_{t=1}^{T_0}\lambda_t Y_{it} - \tfrac{1}{T_1}\!\!\sum_{t=T_0+1}^{T}\! Y_{it}\Big)^2$$`

over `$\Lambda = \{\lambda \geq 0, \sum_t \lambda_t = 1\}$`.

- No statistical ridge here: `$T_0$` is small, no collinearity to fix (a
  negligible numerical regularizer only).
- The "before" baseline is **not** a flat average of all history. It
  concentrates on the pre-periods most predictive of the post-period.

.highlight-box[
On Prop 99, only **3 of 19** pre-periods carry meaningful `$\hat\lambda$`
weight. DiD implicitly uses all 19 equally; that is a modeling choice SDID
declines to make.
]

---

# What the Weights Buy: Local Parallel Trends

DiD assumes parallel trends **globally**: the raw average of all controls
would have moved parallel to the treated units.

SDID weakens this to a **local** condition:

- `$\hat\omega$` builds *one* synthetic control whose pre-trend already matches
  the treated trajectory.
- Parallel trends need only hold for **that** weighted comparison...
- ...over the `$\hat\lambda$`-weighted **periods**, not all of history.

.blue-box[
You are no longer betting that Utah is a good counterfactual for California.
You are betting that a *specific convex combination* of donor states is,
over a *specific weighted window*. That is a far weaker, checkable bet, and
it is why SDID survives the factor structures that sink DiD.
]

---

# The Bridge: One Square, Three Estimators

.small[
| | uniform time weights `$\lambda_t = 1/T$` | optimized time weights `$\hat\lambda$` |
|---|---|---|
| **uniform unit weights** `$\omega_i = 1/N$` | **DiD** | time-weighted DiD |
| **optimized unit weights** `$\hat\omega$` | **SC** (drop unit FE) | **SDID** |
]

- **SDID `$\to$` DiD** when both weight vectors go uniform: the weighted TWFE
  collapses to ordinary TWFE.
- **SDID `$\to$` SC** when the time weights match SC's implicit comparison and
  the unit intercept is switched off (with `$\omega_0 = 0$`, `$\alpha_i \equiv 0$`).

.highlight-box[
DiD and SC are the two axis-aligned corners. SDID is the interior point that
reweights on **both** margins. "Which method?" becomes "which corner am I
willing to sit in, and why?"
]

---

# Why the Intercept Changes Everything

Pure **SC** must match *levels*: the synthetic control has to sit on top of
the treated unit throughout the pre-period. If no donor mix reaches that
level, SC is biased.

**SDID** absorbs the level gap into `$\omega_0$` (and `$\alpha_i$`). Donors only
reproduce the treated unit's *trend*. This is the difference-in-differences
move (levels cancel in the double difference) imported into the SC weighting
step.

.pull-left[
.blue-box[
**SC needs a good fit.**
The synthetic must overlap the treated pre-path in level and shape.
]
]

.pull-right[
.highlight-box[
**SDID needs a parallel fit.**
The synthetic must move *parallel* to the treated pre-path. A constant gap
is fine.
]
]

That one degree of freedom is the whole reason SDID needs no perfect
pre-treatment fit, only a parallel one.

<a href="#intercept-detail" class="nav-btn">worked contrast</a>

---

# Live: California Prop 99

The canonical case: a 1988 tobacco tax, one treated state, 38 donors,
31 years.

```r
data("california_prop99", package = "synthdid")
setup <- panel.matrices(california_prop99)   # long -> Y, N0, T0, W
c(N = nrow(setup$Y), T = ncol(setup$Y), N0 = setup$N0, T0 = setup$T0)
```

```
##  N  T N0 T0 
## 39 31 38 19
```

```r
tau_sdid <- synthdid_estimate(setup$Y, setup$N0, setup$T0)
tau_sc   <- sc_estimate(setup$Y, setup$N0, setup$T0)   # same package
tau_did  <- did_estimate(setup$Y, setup$N0, setup$T0)  # uniform weights
```

`panel.matrices()` reshapes the long panel into the outcome matrix `Y` with
treated units in the last rows and post-periods in the last columns, exactly
the block structure the estimator assumes.

---

# DiD vs SC vs SDID, Side by Side

```r
set.seed(1)   # placebo variance resamples control units
imap_dfr(ests, ~ tibble(
  Estimator = .y,
  Estimate  = as.numeric(.x),
  `Placebo SE` = sqrt(vcov(.x, method = "placebo")))) |>
  mutate(across(where(is.numeric), ~ round(.x, 2))) |>
  knitr::kable(format = "html")
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Estimator </th>
   <th style="text-align:right;"> Estimate </th>
   <th style="text-align:right;"> Placebo SE </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> SDID </td>
   <td style="text-align:right;"> -15.60 </td>
   <td style="text-align:right;"> 10.05 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> SC </td>
   <td style="text-align:right;"> -19.62 </td>
   <td style="text-align:right;"> 10.99 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> DID </td>
   <td style="text-align:right;"> -27.35 </td>
   <td style="text-align:right;"> 16.33 </td>
  </tr>
</tbody>
</table>

.highlight-box[
DiD is the most negative and least precise: equal donor weights, global
parallel trends, both violated here. SC reweights donors but pays for the
level match. **SDID sits between them in point estimate and has the smallest
SE**, the AER paper's headline in three rows.
]

---

# The Signature Plot

.small[
Treated (blue) vs synthetic (red) trajectory; the shaded band marks the
`$\hat\lambda$`-weighted pre-periods; the arrow/parallelogram side is
`$\hat\tau$`. SDID's synthetic runs *parallel* to the treated pre-path with a
gap; DiD's is a flat level shift; SC's overlaps.
]

---

# Reading the Unit Weights

.small[
Each donor's post-period contribution, sized by its `$\hat\omega$` weight. A
few states (Nevada, New Hampshire, Connecticut) carry most of the mass; the
horizontal line is `$\hat\tau$`. This is the SC transparency SDID inherits:
you can name the counterfactual.
]

---

# Inference: Placebo and Jackknife

Both estimators treat the weights as fixed (their estimation is first-order
negligible).

- Reassign treatment to controls, one at a time; variance of the placebo
  effects.
- **Works with one treated unit** (the usual SC case).
- Default on Prop 99.
]

- Delete one unit, re-estimate; usual jackknife variance.
- Cheaper, lighter assumptions.
- **Needs `$\geq 2$` treated units**: with one, it is undefined.
]

```r
set.seed(1)
c(placebo   = sqrt(vcov(tau_sdid, method = "placebo")),
  jackknife = sqrt(vcov(tau_sdid, method = "jackknife")))  # NA: 1 treated unit
```

```
##   placebo jackknife 
##  10.05324        NA
```

<a href="#jackknife-detail" class="nav-btn">when to use which</a>

---

# Application 2: A Panel Where DiD Must Fail

One treated city, 20 donors, 48 months. Outcomes driven by **two latent
factors** with city-specific loadings, plus an airport-pickup policy at
month 36 (true effect `$= 4.0$`).

---

# The Same Problem at an Online Retailer

- A handful of metros receive next-day delivery upgrades in the same quarter; this is the multi-treated-unit case where synthetic control's single-unit logic no longer applies directly.
- SDID reweights the donor metros (those still on two-day) to match the pre-launch order trend of the treated group, relaxing the strict parallel-trends assumption that DiD requires.
- It also downweights remote pre-periods where common-trend assumptions are least credible, keeping identification anchored on the recent pre-launch window.
- The unit weights reveal which donor metros anchor the counterfactual; metros with similar pre-launch order trajectories and demand density receive the most weight.
- SDID sits between DiD (fully trusting parallel trends) and SC (demanding perfect pre-period fit), and is preferred when treated metros are few but more than one.

---

# Factor Structure: SDID Recovers, DiD Does Not

```r
set.seed(1)   # placebo variance resampling
imap_dfr(rs_ests, ~ tibble(
  Estimator = .y, Estimate = as.numeric(.x),
  `Placebo SE` = sqrt(vcov(.x, method = "placebo")),
  `|error|`  = abs(as.numeric(.x) - 4.0))) |>       # truth = 4.0
  mutate(across(where(is.numeric), ~ round(.x, 2))) |>
  knitr::kable(format = "html")
```

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Estimator </th>
   <th style="text-align:right;"> Estimate </th>
   <th style="text-align:right;"> Placebo SE </th>
   <th style="text-align:right;"> |error| </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> SDID </td>
   <td style="text-align:right;"> 3.94 </td>
   <td style="text-align:right;"> 0.61 </td>
   <td style="text-align:right;"> 0.06 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> SC </td>
   <td style="text-align:right;"> 4.31 </td>
   <td style="text-align:right;"> 1.54 </td>
   <td style="text-align:right;"> 0.31 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> DID </td>
   <td style="text-align:right;"> 6.29 </td>
   <td style="text-align:right;"> 1.75 </td>
   <td style="text-align:right;"> 2.29 </td>
  </tr>
</tbody>
</table>

.highlight-box[
DiD reads **more than half the trending-factor divergence as treatment
effect** (6.29 vs 4.0). SC and SDID reweight toward the high-loading donor
cities and recover the truth; SDID is the sharpest. This is the
interactive-fixed-effects regime where SDID and `gsynth` belong and DiD does
not.
]

.tiny[
Same shape, different label: a fee change shipped in one country of a
multi-country marketplace, or a policy applied to one product category.
One treated unit, factor structure in the donors, same estimator.
]

---

# When to Reach for SDID

.small[
| Situation | Call |
|---|---|
| Panel, clear pre-period, few treated units, suspected non-parallel controls | **SDID** |
| Unit weights near uniform, time weights near flat | DiD is fine; SDID confirms it |
| Perfect pre-fit achievable, level match matters | SC still reasonable |
| Interactive fixed effects / factor structure | SDID or `gsynth` (Module 4) |
| One treated unit | placebo SE |
| Several treated units | jackknife SE |
| Staggered adoption / irregular missingness | combine with Modules 1-2, or matrix completion (Module 8) |
]

.blue-box[
The interview one-liner: "SDID is a weighted DiD whose unit weights are a
regularized synthetic control with an intercept, and whose time weights pick
the comparison periods. It degrades to DiD when weights go uniform and to SC
when you drop the intercept, and it is doubly robust: consistent if either
weight set is right."
]

---

# Interview Questions

.small[
| Question | Core of a strong answer |
|---|---|
| "How is SDID different from synthetic control?" | Adds time weights and a unit intercept/FE, plus ridge on the unit weights. The intercept means donors match trend not level, so no perfect pre-fit is needed. |
| "When does SDID reduce to plain DiD?" | When unit and time weights are both uniform: the weighted TWFE becomes ordinary TWFE. Happens when donors are exchangeable and history is homogeneous. |
| "Why is SDID often more precise than SC or DiD?" | Time weights concentrate the comparison on informative periods and the ridge spreads unit weight across donors, both reducing variance. On Prop 99 its SE is the smallest of the three. |
| "You have one treated market. How do you get a standard error?" | Placebo variance: reassign treatment to controls, take the spread of placebo effects. Jackknife is undefined with one treated unit. |
| "What is SDID's identifying assumption?" | Local parallel trends: the `$\hat\omega$`-weighted control moves parallel to the treated unit over the `$\hat\lambda$`-weighted pre-periods. Weaker than DiD's global PT. |
| "Why did the number change when you switched from synthetic control to synthetic DiD?" | Different assumptions: SC matches levels, SDID matches trends with a level shift absorbed by the intercept. If the two agree, that is reassurance; if they diverge, the level match was doing work, and you should say which assumption you believe. |
]

---

# Going Deeper

.small[
| Paper | What it adds |
|---|---|
| Arkhangelsky, Athey, Hirshberg, Imbens and Wager (2021), *AER* | The estimator on these slides: dual weights, `$\zeta$`, asymptotics, placebo/jackknife inference. |
| Abadie, Diamond and Hainmueller (2010), *JASA* | The SC parent: unit weights matching pre-period levels, placebo-in-space inference. |
| Doudchenko and Imbens (2016), *NBER WP* | The synthesis that anticipated SDID: intercept + unconstrained weights bridge DiD and SC. |
| Xu (2017), *Political Analysis* | Generalized SC / interactive fixed effects: the factor-model view SDID approximates. |
| Clarke, Pailañir, Athey and Imbens (2023) | Practical `synthdid` guidance: staggered adoption, covariates, inference choices. |
]

**Next module:** we leave panels for cross-sections and heterogeneous
effects: causal forests, honest splitting, and pointwise CIs.

**Drill:** `exercise.R` reconstructs `$\hat\tau$` from the package's own
weights to 1e-10, then re-solves both weight programs with `quadprog` and
compares to the package's Frank-Wolfe solution.

---

# Backup Slides

---

# Backup: SDID Is a Weighted Double Difference

.small[
Given the weights, extend `$\hat\omega$` with `$1/N_1$` on each treated unit and
`$\hat\lambda$` with `$1/T_1$` on each post-period. Then `$\hat\tau$` is the
weighted double difference:

`$$\hat\tau = \Big(\underbrace{\bar Y^{\text{post}}_{\text{tr}}}_{\text{treated, post}} - \underbrace{\textstyle\sum_t \hat\lambda_t \bar Y^{t}_{\text{tr}}}_{\text{treated, }\hat\lambda\text{-pre}}\Big) - \Big(\underbrace{\textstyle\sum_i \hat\omega_i \bar Y^{\text{post}}_{i}}_{\hat\omega\text{-ctrl, post}} - \underbrace{\textstyle\sum_i\sum_t \hat\omega_i \hat\lambda_t Y_{it}}_{\hat\omega,\hat\lambda\text{-ctrl, pre}}\Big)$$`

- First parenthesis: treated units' post-minus-(weighted)-pre change.
- Second: the synthetic control's post-minus-(weighted)-pre change.
- Their difference is a DiD, but with donor weights `$\hat\omega$` **and**
  period weights `$\hat\lambda$` instead of uniform ones.

Set `$\hat\omega_i = 1/N_0$` and `$\hat\lambda_t = 1/T_0$` and this is exactly
the two-group DiD formula from Module 1. The exercise verifies the identity
holds to 1e-10 using `attr(tau, "weights")`.
]

---

# Backup: Solving the Unit Program with quadprog

.small[
The unit program is a quadratic program. Stack the intercept and weights as
`$b = (\omega_0, \omega)$`, let `$X$` be the `$T_0 \times (N_0 + 1)$` matrix
`$[\,\mathbf{1}\ \ Y^{\text{ctrl, pre}\top}\,]$` and `$y$` the treated
pre-period average. Minimize

`$$\lVert Xb - y\rVert^2 + \zeta^2 T_0\, b^\top P b,
\qquad P = \mathrm{diag}(0, 1, \dots, 1)$$`

(the `$0$` leaves the intercept unpenalized), subject to
`$\sum_i \omega_i = 1$` and `$\omega_i \geq 0$`. In `quadprog` form,
`$D = 2(X^\top X + \zeta^2 T_0 P)$`, `$d = 2 X^\top y$`, one equality row and
`$N_0$` non-negativity rows:

```r
Dmat <- 2 * (t(X) %*% X + zeta^2 * T0 * P)
Amat <- cbind(c(0, rep(1, N0)), rbind(0, diag(N0)))  # sum=1, then w>=0
solve.QP(Dmat, 2 * t(X) %*% y, Amat, c(1, rep(0, N0)), meq = 1)
```

The package uses **Frank-Wolfe**, not a QP solver, so the two agree closely
but not exactly. On Prop 99: `$\mathrm{cor}(\hat\omega_{\text{QP}}, \hat\omega_{\text{pkg}}) \approx 0.9998$`
and the two `$\hat\tau$` differ by `$\approx 4\times 10^{-4}$`. Close, and the exercise says so out loud.
]

---

# Backup: The Regularization Parameter

.small[
`$$\zeta = (N_1 T_1)^{1/4}\, \hat\sigma,
\qquad
\hat\sigma^2 = \frac{1}{N_0 (T_0 - 1)}
\sum_{i=1}^{N_0}\sum_{t=1}^{T_0 - 1}
\left(\Delta_{it} - \bar\Delta\right)^2,
\quad \Delta_{it} = Y_{i, t+1} - Y_{it}$$`

- `$\hat\sigma$` is the SD of **first-differenced** control outcomes over the
  pre-period: a scale-free estimate of one-period noise, robust to unit
  levels and common trends (differencing kills both).
- `$\bar\Delta$` is the average first difference (the common drift).
- The `$(N_1 T_1)^{1/4}$` factor makes the ridge shrink at the rate that keeps
  the estimator `$\sqrt{N_1 T_1}$`-consistent as the panel grows.

Why penalize at all? Without it, collinear donors let the weights explode to
chase noise in the pre-fit (classic SC overfitting). The ridge spreads mass
and stabilizes. On Prop 99, `$\hat\sigma \approx 5.5$` and `$\zeta \approx 10.2$`;
the exercise computes both from the raw matrix.

Note: the **time** program uses only a negligible regularizer. Time weights
face no donor-collinearity problem (here `$T_0$` is small), so no statistical
shrinkage is needed there.
]

---

# Backup: Intercept, Worked Contrast

.small[
Suppose the treated unit sits 20 units *above* every donor in the
pre-period, but moves perfectly parallel to donor 3.

**Pure SC** (no intercept, weights on the simplex): no convex combination of
donors can reach 20 units above all of them, since a weighted average of the
donors lies *within* their range. SC settles for the closest reachable
level, distorting the weights away from donor 3 and biasing the
counterfactual.

**SDID** (intercept `$\omega_0$`): set `$\omega_0 = 20$` and put all weight on
donor 3. The pre-fit is exact *in shape*; the 20-unit level gap is carried
by the intercept, and later by the unit fixed effect `$\alpha_i$` in the
weighted regression. Donor 3 is correctly identified as the counterfactual.

This is the difference-in-differences insight (a constant level gap is
harmless because it cancels in the double difference) transplanted into the
weighting step. It is also why SDID does not require, and does not reward,
a perfect-level pre-fit the way SC does.
]

---

# Backup: Placebo vs Jackknife, Concretely

.small[
| | Placebo | Jackknife |
|---|---|---|
| Idea | assign fake treatment to controls | leave-one-unit-out resampling |
| Treated units needed | `$\geq 1$` | `$\geq 2$` |
| Assumption | homoskedastic-style across units | lighter, standard jackknife |
| Cost | one re-estimate per control | one re-estimate per unit |
| Prop 99 | the default (1 treated state) | returns `NA` |

**Why jackknife breaks with one treated unit.** The jackknife variance sums
squared deviations of leave-one-out estimates. Deleting the single treated
unit removes *all* treated variation, so that leave-one-out estimate is
undefined; the package returns `NA` rather than a misleading number.

**Placebo mechanics.** For each control `$j$`, pretend `$j$` was treated at the
same time, re-estimate on the remaining controls (true effect zero), collect
`$\hat\tau^{\text{placebo}}_j$`. The variance of these is the placebo variance.
It is exact under the assumption that the treated unit is exchangeable with
the controls in its noise distribution.

**Rule.** One treated unit: placebo. Several treated units: jackknife
(cheaper, and the exchangeability assumption is doing less work). Many
treated units: bootstrap is also available.
]

---

# Backup: The Factor-Structure DGP

```r
make_rideshare <- function(n_donors = 20, n_t = 48, g = 36, seed = 7) {
  set.seed(seed)
  N  <- n_donors + 1                                    # city 1 = treated
  f1 <- seq(0, 6, length.out = n_t) + cumsum(rnorm(n_t, 0, 0.3))    # trend
  f2 <- 2 * sin(2 * pi * (1:n_t) / 12) + cumsum(rnorm(n_t, 0, 0.2)) # seasonal
  L1 <- c(1.5, runif(n_donors, 0.2, 1.7))              # treated loads HIGH on f1
  L2 <- c(1.0, runif(n_donors, 0.2, 1.6))
  mu <- c(30, runif(n_donors, 20, 40))                 # city levels
  expand_grid(city = 1:N, t = 1:n_t) |>
    mutate(post = t >= g, D = (city == 1) & post,
           y = mu[city] + L1[city]*f1[t] + L2[city]*f2[t] +
               if_else(D, 4.0, 0) + rnorm(n(), 0, 1))   # true effect = 4
}
```

- Two latent factors with **city-specific loadings** is exactly an
  interactive-fixed-effects model: `$Y_{it} = \mu_i + L_i^\top f_t + \tau D_{it} + \varepsilon_{it}$`.
  Parallel trends fails because loadings differ across cities.
- The treated city's high `$f_1$` loading (1.5, near the top of the donor
  range) makes its counterfactual *diverge* from the donor average, so DiD
  is biased upward. SC and SDID reweight toward high-loading donors and
  recover the truth.
- The treated loading stays *inside* the donor hull (donors span 0.2 to
  1.7), so a convex synthetic control exists.
]