Module 4: Synthetic Control

.title[
# Module 4: Synthetic Control
]
.subtitle[
## Estimator, Inference, Augmented and Generalized Variants
]

---

# 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><b>4</b></td><td><b>Synthetic Control: Estimator, Inference, Variants</b> <i>(you are here)</i></td><td>current</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 tour showed synthetic control as a convex combination of donors fit by
`solve.QP`. This module gives the estimator formally and the two variants a
practitioner actually chooses between.

**You will be able to:**

1. Write the Abadie-Diamond-Hainmueller estimator, including the nested V-matrix
   optimization, and say why the no-intercept and simplex constraints matter.
2. Run placebo-in-space with the RMSPE-ratio statistic and compute an exact
   permutation p-value; run placebo-in-time; describe conformal SC.
3. Explain augmented SC (bias correction) and generalized SC (interactive fixed
   effects), and pick between SC, ASCM, and gsynth for a given panel.

**Applications:** California Prop 99 (canonical, built from scratch and checked
against `Synth`), then a ride-share single-city policy on a simulated factor
panel where `gsynth` and `augsynth` earn their keep.

---

# When You Have One Treated Unit

California passed Proposition 99 in 1989 (cigarette tax plus anti-smoking
programs). No other state did the same thing at the same time. Build a
counterfactual California from the 38 states that did not.

No single state matches California. A convex combination of several might.

---

# The Abadie-Diamond-Hainmueller Estimator

Unit 1 is treated, units `$2,\dots,J{+}1$` are donors. Stack `$k$` predictors into
`$X_1$` (treated, `$k \times 1$`) and `$X_0$` (donors, `$k \times J$`). Choose weights

`$$W^*(V) = \arg\min_{W}\; (X_1 - X_0 W)'\, V\, (X_1 - X_0 W)
\quad \text{s.t.}\quad w_j \ge 0,\; \textstyle\sum_j w_j = 1.$$`

- `$w_j \ge 0$` and `$\sum w_j = 1$`: the synthetic unit is a weighted average, so
  it lives *inside the convex hull* of the donors. No extrapolation.
]

Unlike DiD or regression, SC has no additive shift. Donors must reproduce the
*level* of the treated unit. That is why pre-fit is the load-bearing diagnostic.
]

.highlight-box[
`$V$` is a diagonal matrix of predictor importances. Fix `$V = I$` and this is one
quadratic program. Optimize `$V$` and it becomes a nested program (next slide).
]

<a href="#adh-qp" class="nav-btn">QP form</a>
<a href="#adh-derivation" class="nav-btn-br">the solve.QP mapping</a>

---

# The V-Matrix: A Nested Optimization

The inner problem gives `$W^*(V)$` for a fixed `$V$`. The outer problem chooses `$V$`
to minimize the pre-treatment fit of the *outcome*:

`$$V^* = \arg\min_{V}\; \big(Z_1 - Z_0\, W^*(V)\big)'\big(Z_1 - Z_0\, W^*(V)\big)$$`

where `$Z_1, Z_0$` are pre-period outcome paths. Every candidate `$V$` requires
re-solving the inner QP: a bilevel program.

.blue-box[
**Why bother?** `$V$` decides which predictors the match prioritizes. With
predictors on different scales (income in dollars, prices in cents, a few
outcome lags), equal weighting is arbitrary and `$V^*$` matters a lot.
]

.small[
When the predictors are *only* outcome lags, all on one scale, the optimal `$V^*$`
is near a scaled identity. We will see this directly on Prop 99: the hand-rolled
`$V = I$` fit and the `Synth` fit that optimizes `$V$` are nearly identical.
]

<a href="#vmatrix-detail" class="nav-btn">nested-optimization detail</a>

---

# Interpolation, Not Extrapolation

The simplex constraint means the synthetic unit can only reach outcome levels
*inside* the convex hull of the donors.

.pull-left[
**If the treated unit is interior** to the hull, SC fits it well and the
counterfactual is a genuine interpolation.
]

.pull-right[
**If the treated unit is extreme** (outside the hull), no convex combination
reaches it. Pre-fit is imperfect *by construction*, and the residual imbalance
biases the estimate.
]

.highlight-box[
SC's refusal to extrapolate is a feature: a bad pre-fit is an honest signal the
donor pool cannot reconstruct the treated unit. Augmented SC (later) buys a bias
correction by *allowing* controlled extrapolation, trading the guarantee for
lower bias.
]

<a href="#hull-detail" class="nav-btn">convex-hull picture</a>

---

# Prop 99 From Scratch

`$V = I$` on the 19 pre-period outcome lags; one quadratic program.

```r
A <- Y0[pre_idx, ]; b <- y1[pre_idx]
J <- ncol(Y0)
# min ||A w - b||^2  s.t. w >= 0, sum w = 1
Dmat <- 2 * (t(A) %*% A + 1e-4 * diag(J))
dvec <- 2 * as.numeric(t(A) %*% b)
Amat <- cbind(rep(1, J), diag(J))
bvec <- c(1, rep(0, J))
w <- solve.QP(Dmat, dvec, Amat, bvec,
              meq = 1)$solution
```
]
]

.pull-right[
<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Donor </th>
   <th style="text-align:right;"> Weight </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Utah </td>
   <td style="text-align:right;"> 0.394 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Montana </td>
   <td style="text-align:right;"> 0.232 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Nevada </td>
   <td style="text-align:right;"> 0.205 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Connecticut </td>
   <td style="text-align:right;"> 0.109 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> New Hampshire </td>
   <td style="text-align:right;"> 0.045 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Colorado </td>
   <td style="text-align:right;"> 0.015 </td>
  </tr>
</tbody>
</table>

.small[
Pre-RMSPE 1.66 packs against a pre-period mean near 110:
an excellent fit. Five donors carry all the weight.
]
]

<a href="#prop99-dgp" class="nav-btn">reshape code</a>

---

# Observed vs Synthetic California

The two paths track tightly through 1988, then California falls away. The gap is
the estimated effect.

---

# The Gap and the ATT

Average post-1989 effect: **-19.5 packs per capita**, widening
over time as the tax and programs compound. Pre-period gap hovers at zero, which
is exactly what the weights were fit to do.

---

# Does Optimizing V Change the Answer?

`Synth::synth()` runs the full nested V-optimization over the same 19 outcome
lags as `special.predictors`.

<table>
 <thead>
  <tr>
   <th style="text-align:left;"> Donor </th>
   <th style="text-align:right;"> V = I (scratch) </th>
   <th style="text-align:right;"> Synth (optimized V) </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;"> Utah </td>
   <td style="text-align:right;"> 0.394 </td>
   <td style="text-align:right;"> 0.396 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Montana </td>
   <td style="text-align:right;"> 0.232 </td>
   <td style="text-align:right;"> 0.229 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Nevada </td>
   <td style="text-align:right;"> 0.205 </td>
   <td style="text-align:right;"> 0.204 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Connecticut </td>
   <td style="text-align:right;"> 0.109 </td>
   <td style="text-align:right;"> 0.109 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> New Hampshire </td>
   <td style="text-align:right;"> 0.045 </td>
   <td style="text-align:right;"> 0.046 </td>
  </tr>
  <tr>
   <td style="text-align:left;"> Colorado </td>
   <td style="text-align:right;"> 0.015 </td>
   <td style="text-align:right;"> 0.015 </td>
  </tr>
</tbody>
</table>

.highlight-box[
Weights agree to within 0.003; the two gap series correlate **1**.
With outcome-lag predictors on one scale, the optimized `$V^*$` lands near the
identity, so the nested search barely moves the estimate. V-optimization earns
its keep only with heterogeneous predictors.
]

<a href="#synth-dataprep" class="nav-btn">dataprep code</a>

---

# Choosing the Donor Pool

The pool is an assumption, not a convenience sample.

.small[
| Criterion | Why | Prop 99 example |
|---|---|---|
| Same outcome-driving structure | Lets the pre-match extend to a post counterfactual | states with similar smoking dynamics |
| No spillovers | A contaminated donor biases the counterfactual | exclude states with cross-border cigarette runs |
| No concurrent shocks | A donor with its own 1990 tax change is not clean | drop states that passed their own controls |
| Screen bad pre-fitters | Poorly-fit donors add noise to placebo inference | trim on pre-RMSPE |
]

.blue-box[
Abadie's discipline: a *small, defensible* pool beats a large one. Every donor
you add is a claim that it shares the treated unit's untreated trajectory. Report
sensitivity to dropping high-weight donors.
]

---

# The Pre-Fit Rule

`$$\text{RMSPE}_{\text{pre}} = \sqrt{\tfrac{1}{T_0}\textstyle\sum_{t \le T_0}
\big(Y_{1t} - \sum_j \hat w_j Y_{jt}\big)^2}$$`

is exactly the objective the weights minimize, so it is the best the pool can do.

.highlight-box[
**Abadie's rule:** if `$\text{RMSPE}_{\text{pre}}$` is large relative to the
treated unit's pre-period outcome scale, do not report the SC estimate. The
synthetic is not tracking the treated unit, so any post-period gap could be fit
error spilling forward rather than a treatment effect.
]

Fixes when the pre-fit is bad: expand the donor pool, lengthen the pre-period,
switch to synthetic DiD (Module 5, which allows an intercept shift), or move to a
different design. On Prop 99 the pre-RMSPE is 1.66, far
below the outcome scale: the fit is trustworthy.

---

# Inference Without Standard Errors

One treated unit means no sampling distribution. Use permutation:
**placebo-in-space**. Re-run SC pretending each donor is treated, against the
rest. For each unit form the ratio

`$$r_i = \frac{\text{RMSPE}_{\text{post},i}}{\text{RMSPE}_{\text{pre},i}}$$`

which normalizes post-period divergence by pre-period fit quality.

The exact permutation p-value is the treated unit's rank in the ratio
distribution:

`$$p = \frac{\#\{i : r_i \ge r_{\text{treated}}\}}{J+1}.$$`

.blue-box[
The *ratio*, not the raw gap, is the statistic: a donor that fits poorly
pre-period will diverge post-period for reasons unrelated to treatment. Dividing
by the pre-RMSPE penalizes exactly that.
]

<a href="#placebo-dgp" class="nav-btn">placebo loop code</a>

---

# Placebo-in-Space: California vs the Cloud

California (red) ratio = **12.4**, rank **3 of 39**,
exact **p = 0.077**. Two never-treated states rank higher: both
fit almost perfectly pre-period, then drift by chance, a reminder that a moderate
pool gives the test limited power.

---

# Trimming Bad Pre-Fitters

Abadie screens placebos whose pre-RMSPE is much larger than the treated unit's:
a state the pool cannot fit is not a fair comparison.

<table>
 <thead>
  <tr>
   <th style="text-align:right;"> States </th>
   <th style="text-align:right;"> Kept (pre-RMSPE &lt;= 2x CA) </th>
   <th style="text-align:right;"> Dropped </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:right;"> 39 </td>
   <td style="text-align:right;"> 29 </td>
   <td style="text-align:right;"> 10 </td>
  </tr>
</tbody>
</table>

.highlight-box[
Here trimming does not change California's rank: the two states above it
(Missouri, Virginia) are themselves *well* fit, so the screen only removes
low-ratio, poorly-fit states. The famous rank-one Prop 99 result uses economic
covariates and an optimized `$V$`, not outcome lags alone. Same method, different
predictor set, different power.
]

---

# Placebo-in-Time (Backdating)

Assign a *fake* treatment date inside the pre-period, refit on data before it,
and inspect the gap up to the real date. A large pseudo-effect undermines the
design.

Fitting on 1970-1979, the 1980-1988 pseudo-effect is -3.4 packs,
about one-sixth of the real -20.8. The big divergence appears only
after 1989.

---

# Conformal Inference (Conceptual)

Chernozhukov, Wuthrich and Zhu (2021) give a model-free alternative to
placebo permutation.

.blue-box[
**The idea.** Under a sharp null of no effect, the post-period SC residuals
should be *exchangeable* with the pre-period residuals. Permute (or block-permute)
residuals across time, and invert the test to a confidence interval for the
entire effect path.
]

- Delivers pointwise and joint confidence bands with finite-sample validity
  under the exchangeability assumption.
- Does **not** need a large donor pool, unlike placebo-in-space.
- The method to reach for when the pool is too small for a credible permutation
  distribution. Implemented in `scinference`.

We describe it; the exercise uses placebo-in-space, which the Prop 99 pool
supports.

<a href="#conformal-detail" class="nav-btn">exchangeability detail</a>

---

# Augmented Synthetic Control

Ben-Michael, Feller and Rothstein (2021) attack SC's residual imbalance
directly. When SC cannot fit perfectly, the leftover
`$X_1 - X_0 \hat W^{\text{scm}}$` biases the estimate. Correct it with an outcome
model:

`$$\hat Y_{1T}(0) = \sum_j \hat w_j^{\text{scm}} Y_{jT} + \big(X_1 - X_0 \hat W^{\text{scm}}\big)'\, \hat\eta$$`

where `$\hat\eta$` is a ridge regression of donor post-outcomes on donor
pre-predictors.

.pull-left[
.highlight-box[
**Perfect pre-fit** makes the imbalance zero, so ASCM `$=$` SC. The correction only
fires when SC leaves imbalance on the table.
]
]

.pull-right[
.blue-box[
**It permits extrapolation.** The correction can push implied weights outside
the simplex (negative weights), reaching a treated unit outside the hull. Ridge
controls how much.
]
]

<a href="#ascm-detail" class="nav-btn">estimator detail</a>

---

# Generalized SC: Interactive Fixed Effects

Xu (2017) embeds SC in a factor model:

`$$Y_{it} = x_{it}'\beta + \lambda_i' f_t + \alpha_i + \xi_t + \varepsilon_{it}$$`

Two-way FE assumes the only time structure is a common shock `$\xi_t$`. The
interactive term `$\lambda_i' f_t$` lets units load differently on `$r$` latent
factors (a demand shock that hits dense cities harder).

.small[
1. Estimate `$\beta$`, factors `$F$`, and control loadings on the **control units**
   only (Bai 2009 IFE estimator).
2. Estimate each treated unit's loadings `$\hat\lambda_i$` from its **pre-treatment**
   residuals on `$\hat F$`.
3. Impute `$\hat Y_{it}(0) = x_{it}'\hat\beta + \hat\lambda_i'\hat f_t$`; average
   `$Y_{it} - \hat Y_{it}(0)$` over treated post cells.
4. Choose `$r$` by cross-validation; bootstrap for inference.
]

.highlight-box[
gsynth generalizes SC: instead of matching factor loadings with convex weights,
it estimates the loadings directly. It handles multiple treated units,
covariates, and extrapolation.
]

<a href="#gsynth-detail" class="nav-btn">estimation detail</a>

---

# Application 2: A Ride-Share Factor Panel

One treated city, 20 donor cities, 48 months. Outcomes are driven by two latent
factors with city-specific loadings; an airport-pickup policy hits the treated
city at month 36 with a true effect of **1.5**. The treated city's loadings
sit near the *edge* of the donor hull, so plain SC will struggle. Outcome: a
rides index (read: units sold or sessions per market in a retail setting).

<a href="#rideshare-dgp" class="nav-btn">factor DGP code</a>

---

# The Same Problem at an Online Retailer

- A single flagship metro receives same-day delivery as a pilot; all other metros remain on two-day and form the donor pool for a synthetic control.
- Pre-period fit minimizes the distance between the flagship's orders-per-capita series and a convex combination of donors, anchoring the synthetic counterfactual to observed pre-launch behavior.
- Placebo-in-space: assign same-day status to each donor metro in turn; the flagship's post-launch gap is significant if it lies outside the distribution of placebo gaps.
- The convex-hull constraint is a real concern if the flagship is the largest market by order volume: no convex combination of smaller donors can match its scale, and the estimator extrapolates rather than interpolates.
- Pre-launch order levels are the natural balancing variables; donor metros whose order trajectories cannot match the flagship's scale receive near-zero synthetic-control weight.

---

# From Policy Evaluation to Geo Experiments

In industry, SC and augmented SC power **geo-level incrementality
measurement** wherever user-level randomization is impossible: brand
advertising, offline media, anything with shared market-level exposure.
Open-source geo-experiment tooling is built on exactly this machinery.

The flip from this module's setting: here you evaluate a policy that
already happened to one city you did not choose. In a **designed geo
experiment** you choose the treated markets, which turns the one-treated-
unit evaluation problem into a design problem, better donor coverage,
pre-registered inference, the same estimator underneath.

.blue-box[
Same math, different posture: post-hoc policy evaluation asks "what was the
effect, given the city that got treated?"; a geo experiment asks "which
cities should we treat so the SC estimate is as sharp as possible?"
]

---

# gsynth Recovers the Factors

Cross-validation picks **r = 2** (the true number of factors). Estimated
ATT **1.59**, 95% CI
[1.11, 2.07], which covers the
truth of 1.5.

---

# SC vs ASCM vs gsynth vs Truth

.tiny[
| Estimator | ATT | Error vs truth | Why |
|---|---|---|---|
| Plain SC | 3.22 | +1.72 | treated unit outside the hull; convex weights cannot reach it |
| Augmented SC | 1.72 | +0.22 | ridge correction removes most of the imbalance bias |
| gsynth | 1.59 | +0.09 | models the two factors directly; CI covers truth |
]

---

# Which Estimator, When

.small[
| Situation | Reach for | Why |
|---|---|---|
| Good pre-fit, treated unit interior to hull | **Plain SC** | interpretable convex weights, no extrapolation, minimal assumptions |
| Good but imperfect pre-fit | **Augmented SC** | keeps SC interpretability, removes first-order imbalance bias |
| Low-rank panel, long pre-period, treated near/outside hull | **gsynth** | estimates factor loadings directly, extrapolates, handles covariates |
| Multiple treated units, staggered adoption | **gsynth** | native support; SC is one-unit-at-a-time |
| Small donor pool (weak placebo distribution) | **conformal SC** | valid inference without a large permutation cloud |
| Want an intercept shift / robustness to pre-fit | **synthetic DiD** | Module 5 |
]

.blue-box[
Default in interviews: "Plain SC if pre-fit is clean and the unit is interior;
augmented SC to bias-correct imperfect fit; gsynth when the panel is low-rank and
I need to extrapolate or handle multiple treated units."
]

---

# Interview Questions

.small[
| Question | Core of a strong answer |
|---|---|
| "Walk me through the synthetic control estimator." | Convex weights on donors minimizing pre-treatment predictor imbalance under `$w_j \ge 0$`, `$\sum w_j = 1$`, no intercept; `$V$` weights predictors and is chosen by an outer loop minimizing pre-period outcome fit. |
| "Pre-fit RMSPE is large relative to the outcome scale. What now?" | Do not report the SC estimate; the counterfactual is not tracking the treated unit. Expand the pool, lengthen the pre-period, switch to synthetic DiD or gsynth, or change the design. |
| "How do you do inference with one treated unit?" | Placebo-in-space: assign treatment to each donor, form the post/pre RMSPE ratio, exact p = rank/(J+1). Back it with placebo-in-time (backdating). Conformal SC for small pools. |
| "When would you use augmented SC over plain SC?" | Pre-fit is good but imperfect. ASCM adds an outcome-model bias correction for the residual imbalance; equals SC when pre-fit is perfect; allows controlled extrapolation via ridge. |
| "What does gsynth buy you over SC?" | An interactive-fixed-effects model: estimate latent factors on controls, project treated loadings, impute the counterfactual. Handles multiple treated units, covariates, and units outside the donor hull; CV picks the factor count. |
| "Marketing ran a brand campaign in one metro. Can you tell me if it worked, since we obviously could not A/B it?" | Yes: that is the synthetic-control setting, one treated market, a donor pool, placebo inference for the p-value. And next quarter, design it as a geo experiment by choosing the treated markets deliberately instead of evaluating after the fact. |
]

---

# Going Deeper

.small[
| Paper | What it adds |
|---|---|
| Abadie, Diamond and Hainmueller (2010, 2015) | The estimator, the V-matrix, placebo-in-space and placebo-in-time inference on Prop 99. |
| Abadie (2021), *JEL* | Practitioner's guide: donor-pool discipline, when SC is feasible, the no-extrapolation logic. |
| Ben-Michael, Feller and Rothstein (2021), *JASA* | Augmented SC: outcome-model bias correction, the extrapolation-vs-bias tradeoff. |
| Xu (2017), *Political Analysis* | Generalized SC / interactive fixed effects; CV factor selection; multiple treated units. |
| Chernozhukov, Wuthrich and Zhu (2021), *JASA* | Conformal inference for SC: exchangeability-based bands, no large pool needed. |
]

**Next module:** synthetic DiD (Arkhangelsky et al. 2021), which adds *time*
weights and an intercept to bridge SC and DiD.

**Drill:** `exercise.R` builds SC from scratch on Prop 99, matches `Synth` to a
gap correlation of 1.00, and runs the full placebo-in-space and backdating tests.

---

# Backup Slides

---

# Backup: SC as a Quadratic Program

.small[
`solve.QP` minimizes `$\frac{1}{2} w' D w - d' w$` subject to `$A' w \ge b$`, with the
first `meq` constraints treated as equalities. Map the SC inner problem
(with `$V = I$`) onto it:

`$$\min_w \|A w - b\|^2 = \min_w\; w'(A'A)w - 2(A'b)'w
\;\Rightarrow\; D = 2 A'A,\;\; d = 2 A'b$$`

```r
A <- Y0[pre_idx, ]; b <- y1[pre_idx]; J <- ncol(Y0)
Dmat <- 2 * (t(A) %*% A + 1e-4 * diag(J))  # ridge -> strictly PD
dvec <- 2 * as.numeric(t(A) %*% b)
Amat <- cbind(rep(1, J), diag(J))          # col 1: sum w = 1 (equality)
bvec <- c(1, rep(0, J))                     # cols 2..: w_j >= 0
w <- solve.QP(Dmat, dvec, Amat, bvec, meq = 1)$solution
```

With 38 donors and 19 pre-years, `$A'A$` is rank-deficient (rank `$\le 19$`), so
`solve.QP`'s Cholesky needs a small ridge to make `$D$` positive definite and the
solution unique and deterministic. The simplex constraints do the real
regularizing: the optimum is a sparse vertex, five non-zero weights here.
]

---

# Backup: Where the QP Comes From

`$$\min_W\; (X_1 - X_0 W)' V (X_1 - X_0 W)
= \min_W\; W'(X_0' V X_0)W - 2(X_0' V X_1)'W + \text{const}.$$`

Matching to `solve.QP`'s `$\frac{1}{2}w'Dw - d'w$` gives

`$$D = 2\, X_0' V X_0, \qquad d = 2\, X_0' V X_1.$$`

Setting `$V = I$` and using the pre-period outcome lags as the predictors, so that
`$X_0$` is the donor pre-period matrix and `$X_1$` the treated pre-period path,
recovers the code on the previous backup. The equality constraint `$\mathbf{1}'W = 1$` is row one of `Amat`
with `meq = 1`; the `$J$` non-negativity constraints `$w_j \ge 0$` are the identity
block. `Synth` replaces the fixed `$V$` with the nested search and uses an
interior-point solver (`ipop` / `LowRankQP`) that tolerates the rank deficiency
without an explicit ridge.
]

---

# Backup: The Nested Optimization

`$$\underbrace{V^* = \arg\min_V \|Z_1 - Z_0 W^*(V)\|^2}_{\text{outer: fit the outcome}}
\quad\text{where}\quad
\underbrace{W^*(V) = \arg\min_{W \in \Delta} (X_1 - X_0 W)'V(X_1 - X_0 W)}_{\text{inner: fit the predictors}}$$`

with `$\Delta$` the simplex. Each outer evaluation solves the inner QP. `Synth`
optimizes `$V$` over the space of diagonal PSD matrices (normalized so predictors
enter comparably) by a derivative-free search (Nelder-Mead / `optim`), starting
from a regression-based or equal-weight `$V$`.

**Why it can matter.** `$V$` is a similarity metric on predictors. If income
(dollars) and price (cents) enter with equal weight, the dollar-scale predictor
dominates the match mechanically. `$V^*$` rescales so the *outcome-relevant*
predictors drive the weights.

**Why it barely moved on Prop 99.** The predictors were all the same variable
(packs) at different years, already on one scale, so any reasonable `$V$` (including
`$I$`) gives nearly the same `$W$`. The `solution.v` from `Synth` is close to uniform
across the 19 lags. This is the exception, not the rule: with genuine covariates,
skip `$V = I$`.
]

---

# Backup: The Convex Hull, Concretely

.small[
Think of each unit's pre-period outcome path as a point in `$T_0$` dimensions.
The synthetic control is a convex combination of the donor points, so it
can only land inside their convex hull.

- **Treated unit interior:** some convex combination reaches it, pre-RMSPE can be
  near zero, and the same weights plausibly reconstruct the untreated
  counterfactual. This is the good case.
- **Treated unit on a vertex / outside:** the closest point in the hull is on its
  boundary, pre-RMSPE is bounded away from zero, and SC systematically over- or
  under-shoots. California smoked *less* per capita than almost every donor even
  pre-1989, which is why the fit, though good, is not perfect.

Two consequences ADH stress:

1. **No extrapolation** means SC never predicts an outcome no donor exhibited: a
   safety property regression lacks.
2. **Interpolation bias** is the price: when the unit is extreme, the convex
   constraint injects bias that augmented SC (allowing weights off the simplex)
   or gsynth (modeling the loadings) can remove.
]

---

# Backup: Reshaping Prop 99

```r
data("california_prop99", package = "synthdid")
cp <- california_prop99 |> mutate(State = as.character(State))

# wide: rows = years, columns = states (a T x N outcome matrix)
wide <- cp |> select(State, Year, PacksPerCapita) |>
  pivot_wider(names_from = State, values_from = PacksPerCapita) |>
  arrange(Year)
years    <- wide$Year
Ymat     <- as.matrix(wide[, -1])       # 31 years x 39 states
pre_idx  <- which(years <= 1988)         # 19 pre-treatment years
post_idx <- which(years >  1988)         # 12 post years
donors   <- setdiff(colnames(Ymat), "California")   # 38 donors

y1 <- Ymat[, "California"]               # treated outcome path
Y0 <- Ymat[, donors, drop = FALSE]       # donor outcome matrix
```

The whole method runs off this one matrix: `sc_weights(y1, Y0, pre_idx)` for the
fit, and the same call with each column swapped in as treated for the placebo
test.
]

---

# Backup: The Synth dataprep Call

.small[
`Synth` wants a plain `data.frame`, numeric unit ids, and predictors specified as
operations. One `special.predictor` per pre-year makes it match on the same 19
outcome lags as the from-scratch fit.

dp <- dataprep(
  foo = d, dependent = "PacksPerCapita", unit.variable = "id",
  time.variable = "Year", unit.names.variable = "State",
  special.predictors = lapply(1970:1988, \(y) list("PacksPerCapita", y, "mean")),
  treatment.identifier = ids$id[ids$State == "California"],
  controls.identifier  = setdiff(ids$id, ids$id[ids$State == "California"]),
  time.predictors.prior = 1970:1988, time.optimize.ssr = 1970:1988,
  time.plot = 1970:2000)
invisible(capture.output(so <- synth(dp, verbose = FALSE)))   # optimizer is chatty
```

`synth()` prints its Nelder-Mead progress; wrap it in `capture.output` and read
`so$solution.w` (donor weights) and `so$solution.v` (predictor importances).
]

---

# Backup: The Placebo-in-Space Loop

```r
placebo_ratio <- function(nm) {
  yy <- Ymat[, nm]
  DD <- Ymat[, setdiff(colnames(Ymat), nm), drop = FALSE]
  w  <- sc_weights(yy, DD, pre_idx)
  g  <- yy - as.numeric(DD %*% w)
  pre_r  <- sqrt(mean(g[pre_idx]^2))     # compute scalars FIRST
  post_r <- sqrt(mean(g[post_idx]^2))
  tibble(state = nm, pre_rmspe = pre_r,
         post_rmspe = post_r, ratio = post_r / pre_r)
}
placebos <- map_dfr(colnames(Ymat), placebo_ratio)
ca_ratio <- placebos$ratio[placebos$state == "California"]
p_value  <- sum(placebos$ratio >= ca_ratio) / nrow(placebos)
```

**A trap worth flagging.** Do *not* write
`tibble(pre = ..., post = sqrt(mean(g[-pre]^2)))`. Inside `tibble()` the new
`pre` column masks the index vector, so `g[-pre]` silently indexes by the scalar
pre-RMSPE and returns the wrong window. Compute the RMSPEs into named scalars
first, then assemble the tibble. Each `solve.QP` is milliseconds, so 39 placebos
run instantly.
]

---

# Backup: Conformal SC, a Little Deeper

`$$\hat u_t = Y_{1t} - \textstyle\sum_j \hat w_j Y_{jt}$$`

Under the sharp null `$H_0: \tau_t = \tau_t^0$` for a hypothesized path, the
adjusted residuals `$\hat u_t - \tau_t^0 \mathbb{1}(t > T_0)$` should be
exchangeable across time.

**The test.** Compute a test statistic (e.g. a block sum of squared post-period
adjusted residuals), then recompute it under all (or many) permutations of the
time index. The conformal p-value is the fraction of permutations with a
statistic at least as large. Invert over `$\tau^0$` to get a confidence set for the
whole effect path or for the average effect.

**Assumptions and payoff.** Exchangeability is weaker than a parametric error
model and does not require many control units, so conformal SC is the inference
tool when the donor pool is too small for a credible placebo-in-space
distribution. The cost: exchangeability can fail under strong serial correlation,
which block permutation partly addresses. Package: `scinference`
(Chernozhukov, Wuthrich and Zhu 2021).
]

---

# Backup: The Augmented SC Estimator

.small[
Write the SC counterfactual as `$\sum_j \hat w_j^{\text{scm}} Y_{jT}$` and its
residual predictor imbalance as `$\delta = X_1 - X_0 \hat W^{\text{scm}}$`. ASCM adds
an outcome-model term:

`$$\hat Y_{1T}(0) = \sum_j \hat w_j^{\text{scm}} Y_{jT} + \delta'\hat\eta,
\qquad
\hat\eta = \arg\min_{\eta}\; \sum_{j} \big(Y_{jT} - X_j'\eta\big)^2 + \zeta\|\eta\|^2$$`

a ridge regression of donor post-outcomes on donor pre-predictors. Equivalently,
ASCM implies augmented weights

`$$\hat w_j^{\text{aug}} = \hat w_j^{\text{scm}} + \big(\text{ridge correction}\big)_j,$$`

which need **not** lie on the simplex: negative weights are allowed, and that is
exactly what lets the estimator reach a treated unit outside the donor hull.

- `$\delta = 0$` (perfect pre-fit) `$\Rightarrow$` correction vanishes, ASCM `$=$` SC.
- Ridge penalty `$\zeta \to \infty$` `$\Rightarrow$` correction shrinks to zero,
  back to SC. `$\zeta \to 0$` `$\Rightarrow$` maximal extrapolation.

`augsynth(..., progfunc = "Ridge", scm = TRUE)` fits it. On the ride-share panel
it cut plain SC's bias from +1.72 to
+0.22.
]

---

# Backup: gsynth Estimation and Inference

`$$Y_{it} = x_{it}'\beta + \lambda_i' f_t + \alpha_i + \xi_t + \varepsilon_{it}$$`

is fit by Xu's (2017) EM-style algorithm:

1. **Controls only.** Iterate between (a) OLS of the FE-demeaned outcome on
   `$x_{it}$` for `$\beta$`, and (b) a principal-components step on the residual matrix
   for `$F$` and control loadings `$\Lambda$` (Bai 2009). Converges to the IFE
   estimates using no treated-unit post data.
2. **Treated loadings.** Regress each treated unit's *pre-treatment* residual
   `$Y_{it} - x_{it}'\hat\beta$` on `$\hat F$` over `$t \le T_0$` to get `$\hat\lambda_i$`.
3. **Impute and average.** Set
   `$\hat Y_{it}(0) = x_{it}'\hat\beta + \hat\lambda_i'\hat f_t$` and take the ATT as
   the mean of `$Y_{it} - \hat Y_{it}(0)$` over treated post cells.

**Factor count.** Chosen by cross-validation: hold out pre-period observations,
fit for each candidate `$r$`, pick the `$r$` minimizing held-out MSPE. On the
ride-share panel CV selected `$r = 2$`, the true value.

**Inference.** Parametric bootstrap resamples the estimated errors (block for
serial correlation), refits, and takes quantiles. Key setup:
`gsynth(y ~ D, index = c("city","month"), force = "two-way", CV = TRUE,
r = c(0, 4), se = TRUE, nboots = 200, inference = "parametric")`.
]

---

# Backup: The Factor-Panel DGP

```r
make_rideshare <- function(seed = 6, n_donor = 20, n_t = 48, t_treat = 36,
                           effect = 1.5, noise = 0.35, treat_shift = 1.2) {
  set.seed(seed); n_units <- n_donor + 1; r <- 2
  F_t <- cbind(cumsum(rnorm(n_t, 0, 0.3)) + sin((1:n_t) / 6),  # latent factor 1
               cumsum(rnorm(n_t, 0, 0.2)))                     # latent factor 2
  L <- matrix(rnorm(n_units * r, 0, 1), n_units, r)            # city loadings
  L[1, ] <- L[1, ] + treat_shift            # treated city near the hull edge
  mu_i <- rnorm(n_units, 8, 1); gamma_t <- 0.02 * (1:n_t)
  units <- tibble(city = 1:n_units, treated_unit = c(TRUE, rep(FALSE, n_donor)))
  expand_grid(city = 1:n_units, month = 1:n_t) |> left_join(units, by = "city") |>
    mutate(post = month >= t_treat, D = as.integer(treated_unit & post),
           y0 = mu_i[city] + gamma_t[month] + rowSums(L[city, ] * F_t[month, ]) +
                rnorm(n(), 0, noise),
           y  = y0 + effect * D)
}
```

The `treat_shift` pushes the treated city's loadings toward the boundary of the
donor-loading cloud, so plain SC (convex weights) cannot reproduce it and
over-shoots, while gsynth (which estimates the loadings) recovers the truth. Two
factors, `$r = 2$`, plus a common trend is a genuinely low-rank panel, exactly the
structure gsynth is built for.
]