11. Synthetic Controls

Causal Data Science for Business Analytics

Christoph Ihl

Hamburg University of Technology

Monday, 6. July 2026

Part V fills the same blanked-out block — each method relaxing parallel trends further.

S10 gave every clean control weight 1.

Today: which controls, and how much?

The cold open frames Part V as one imputation problem in which each method relaxes parallel trends further. Hold the spreadsheet image one more time: a panel of markets and weeks, with a blanked-out rectangle for the never-observed Y(0) of the treated market after launch. Part V is one long project to impute that rectangle under a progressively weaker restriction on trends: S10 filled it with additive unit-plus-time effects (parallel trends — the strongest restriction), S11 relaxes that to a weighted blend of donor markets (synthetic control), S12 replaces it with a flexible factor model. The single sentence to carry through the deck: difference-in-differences is the special case that gives every clean control the same weight, 1; synthetic control asks the next question — which donors, and how much weight on each. That question IS today’s syllabus. Open on the weights, not on a method name.

Intro

The board doesn’t want the average — it wants Northgate

S10 answered “what’s the app worth on average?” The board now asks a different question: what did it do to Northgate — our flagship, market M17, the largest, seven stores, go-live week 53.

Open Act 1 by sharpening the question from a portfolio average to a single unit. S10 delivered the overall ATT — roughly +5% across all treated markets — but a board does not run a portfolio; it runs flagship decisions. The question now is narrow and concrete: what did the loyalty app do to Northgate specifically, market M17, region 4, urban, the single largest market with seven stores, launched in the first wave at week 53 of 104. Point at the figure: Northgate (bold) sits clearly ABOVE the fifteen never-treated donor markets (grey) even before launch — it is the biggest market, at the edge of the pack. That edge position is not a throwaway detail; it is the villain of Act 2, the reason a naive synthetic control will overshoot. Plant it now as a visual fact and let it pay off later. The donor pool is fixed: the fifteen never-treated markets, observed across all 104 weeks.

One treated unit breaks the DiD machinery

DiD needed a cohort — many units launched at similar times, so parallel trends could lean on an average and a comparison pool.

Northgate is one unit. Its pre-trend is mostly noise; there is no cohort to average over.

The single-unit problem

With one treated market and no cohort, “do the treated and controls move in parallel?” has nothing to discipline it. We need a different way to build the counterfactual.

To be clear, the number is not the problem: a plain 2×2 DiD on Northgate already gives +7.7% (true +8.3%). What fails at N = 1 is defending it — no cohort, no clustered inference, no pre-test power.

Name why the S10 apparatus does not transfer. Every robust DiD estimator leaned on cohorts — multiple units treated at similar times — so the group-time ATT averaged over many treated units and parallel trends was disciplined by a whole comparison pool. Northgate is a single unit. A single unit’s pre-period trajectory is mostly idiosyncratic noise, so “are the treated and control trends parallel?” has almost nothing to constrain it: with one treated line, any donor average can be made to look parallel or not depending on which donors you pick. There is no cohort to average over, no way to borrow strength across treated units. This is not a failure of DiD so much as a boundary — the exact boundary S10’s final act named. The constructive response is to stop asking every clean control to play the same role and instead build a tailored counterfactual: a weighted blend of donors chosen to track THIS unit. That is synthetic control: where S10 gave every clean control the same weight of 1, we now ask which donors, and how much weight on each.

A weighted control is a synthetic unit

DiD and synthetic control differ in one choice — the donor weights. That flips the direction the missing \(Y(0)\) is imputed from:

• DiD: every clean control gets weight 1. It projects the treated unit forward in time on the donors’ common trend — a horizontal borrow.
• Synthetic control: solve for a convex blend \(\omega\) (non-negative, summing to 1) that tracks the treated unit pre-launch. Each missing cell is a weighted column of donors — a vertical borrow.

The blend is a synthetic Northgate — a fictional market from real donors, standing in for the one we can’t observe.

State the conceptual core of the method in one comparison. DiD and synthetic control are the same idea — impute Y(0) from controls — differing only in the weights, and that one change flips the DIRECTION of the imputation on the units-by-time panel. DiD gives every clean control the same weight (a simple average) and asks the treated unit to have moved in parallel with that average over TIME — it borrows horizontally, projecting the treated unit’s own pre-period forward on the donors’ common trend. Synthetic control instead SOLVES for the weights: it searches over convex combinations of donors — non-negative, summing to one — whose pre-treatment trajectory tracks the treated unit’s, and imputes each missing post-period cell as a weighted column of donors — it borrows vertically, across units within each period. That weighted blend is a “synthetic” version of the treated unit, a fictional market assembled from real donors; after treatment, when the real treated unit’s Y(0) is unobservable, the synthetic unit keeps tracking forward and serves as the counterfactual. The horizontal-vs-vertical picture is the seed for the “three corners” slide in the Synthetic DiD act (DiD weights time, SC weights units, synthetic DiD weights both) and for the S12 matrix view. The convexity restriction — forbidding negative weights and extrapolation — gets its own slide in the next act; it keeps the synthetic unit inside the range of observed donor behavior.

Building a Synthetic Unit

Match the pre-period, not the covariates alone

Abadie-Diamond-Hainmueller’s objective: choose donor weights \(\omega\) to make the synthetic unit’s pre-treatment outcome path track the treated unit’s. (Abadie, Diamond, and Hainmueller 2010)

\[\min_{\omega}\ \sum_{t \le T_0}\Big(Y^{\text{treat}}_t - \sum_{j}\omega_j\,Y^{\text{donor}}_{j,t}\Big)^2 \quad \text{s.t. } \omega_j \ge 0,\ \textstyle\sum_j \omega_j = 1\]

The constraints make \(\omega\) a point in the simplex — non-negative, summing to 1. Convexity = no extrapolation: the synthetic unit stays inside the donors’ range.

Introduce the ADH estimator as the spine of the act. The objective is to choose donor weights omega to minimize the discrepancy between the treated unit’s pre-treatment outcome path and the weighted donors’ path — match the trajectory, the thing we actually want to extrapolate, not just a list of covariate means. The two constraints define the convex hull, the simplex: weights are non-negative and sum to one. This convexity is the method’s signature design choice and its central limitation. Its virtue: forbidding negative weights and weights above one means the synthetic unit is an INTERPOLATION of observed donors — it never extrapolates beyond the range of behavior any donor actually exhibited, which guards against the wild out-of-sample predictions that unconstrained regression can produce. Its cost, which Act 2 will hit hard: if the treated unit lies OUTSIDE the convex hull of the donors — for instance, if it is the largest market and no blend of smaller donors can match its level — then no convex weighting can fit it, and the residual pre-period misfit becomes bias. Covariates can be added to the matching, but the pre-period outcome path is the dominant target; keep the focus there.

California’s tobacco tax: the textbook fit

Abadie’s original case: California’s Proposition 99 (1988). Synthetic California — a blend of other states — tracks the real state almost exactly before the tax (pre-RMSPE ≈ 1.8 packs). (Abadie, Diamond, and Hainmueller 2010, 2015)

Ground the method in the case that made it famous, because it shows what a GOOD fit looks like before we see a bad one. Abadie, Diamond and Hainmueller studied California’s Proposition 99, the 1988 tobacco tax, asking what happened to per-capita cigarette consumption. They built a synthetic California from a convex blend of other US states chosen to match California’s pre-1988 smoking trajectory and predictors. The fit is the textbook ideal: before 1988 the synthetic line lies almost exactly on top of real California, with a pre-treatment root-mean-squared prediction error of roughly 1.8 packs per capita — small relative to the scale. Why does it fit so well? California, though large, sits comfortably inside the convex hull of the other states on the smoking dimension, so a blend can match it. Contrast this deliberately with Northgate, which does NOT sit inside its donors’ hull. The clean California fit earns the right to read the treatment effect straight off the post-period gap — which is the next slide.

The gap plot reads the treatment effect off

Subtract synthetic from real and plot the gap. Flat at zero before 1988 (the fit was clean); it opens to ≈ −18.7 cigarettes/capita after — the effect of the tax.

Teach the gap plot as the standard synthetic-control read, and emphasize the logic that makes it trustworthy. Plot the difference between the real unit and its synthetic counterpart over time. Before treatment, a good fit means the gap hovers at zero — that flatness is the visual proof the synthetic unit is a faithful stand-in. After treatment, the gap is the estimated treatment effect at each period: for Prop 99 it opens steadily downward to about minus 18.7 cigarettes per capita by the late 1990s, a large and growing reduction in smoking. The crucial inferential point, which the next two slides exploit: the gap plot is only interpretable if the PRE-period gap is flat at zero. A flat pre-gap licenses reading the post-gap as the effect. A non-flat pre-gap means the synthetic unit never matched the treated unit in the first place, so the post-gap mixes the true effect with the pre-existing misfit — and that is precisely the disease Northgate has.

Northgate doesn’t fit as cleanly as California

Same recipe, harder unit. Northgate is the largest market — at the edge of the donor convex hull. No blend of smaller donors reaches its level: the dashed synthetic sits persistently below before launch.

This is the diagnostic moment — show the imperfect fit and name its cause precisely. We run the identical ADH recipe on Northgate, donor pool the fifteen never-treated markets, and the pre-period fit is visibly worse than California’s: the synthetic line sits persistently BELOW real Northgate even before week 53, never quite reaching its level. The reason is the convexity constraint colliding with Northgate’s position. Northgate is the largest market in the portfolio; every donor is smaller. A convex blend — weights summing to one, none negative — can only produce a synthetic unit whose level lies BETWEEN the donors’ levels, so it cannot reach above all of them. Northgate sits at the EDGE of (effectively outside) the convex hull, and no convex weighting can match a point outside the hull. The result is a structural pre-period offset: the synthetic is biased low throughout. This is not a tuning problem you can fix by trying harder within the simplex; it is the simplex itself. The next slide shows what that offset does to the estimate.

An imperfect pre-fit inflates the estimate

The pre-gap isn’t flat — it sits at +6.8% (pre-RMSPE ≈ 0.070). That offset leaks straight into the post-gap:

+15.2% ≈ pre-offset +6.8% + true effect +8.3%

This is the mechanical heart of Act 2 — the single most important arithmetic in the session. Because the synthetic Northgate sits about 6.8% below the real one throughout the PRE-period (root-mean-squared prediction error around 0.070), the gap plot does not start at zero — it starts at +6.8%. That baked-in offset does not vanish at launch; it persists into the post-period and adds to whatever true effect there is. So the post-period gap of +15.2% decomposes almost exactly into the pre-period offset (+6.8%) plus the true Northgate effect (+8.3%): 6.8 + 8.3 = 15.1, rounding to the reported +15.2%. Say this slowly and point at the decomposition: the classic SC overshoots not because the effect is large but because the FIT is bad, and a bad fit contaminates the estimate one-for-one. This is the synthetic-control analog of S10’s contamination story, and it reframes the gap plot’s pre-period as a diagnostic you must check before trusting any number. Everything in Act 4 (augmented SC) is about killing that +6.8% offset.

Read the weights: which markets, how much

The blend is maximally sparse — one donor (M29) carries essentially all the weight, the rest sit at zero. That single weight vector is still a row describing how to rebuild one unit from the others (a seed for the matrix view later).

Make the weights tangible and plant a seed for the matrix view in Act 5. Synthetic-control weights are typically SPARSE — the simplex constraint plus the optimization tend to load most of the weight onto a handful of donors and leave the rest at zero. That sparsity is a feature: it means you can name the markets your synthetic Northgate is actually made of and sanity-check them (are they plausible comparators? do they share Northgate’s region, urbanicity, size?), which is the interpretability that makes synthetic control attractive to a board. Read a couple of the tall bars aloud as “synthetic Northgate is mostly these markets.” Then plant the Act 5 seed quietly: this weight vector is a single ROW of instructions — how to reconstruct one unit’s outcome path from the others. When we get to synthetic DiD and then preview S12, we will stack such rows and recognize them as a low-rank matrix. Do not over-explain it here; just leave the phrase “weights are a row” on the table for later.

Donor hygiene decides what the weights can mean

A synthetic unit is only as good as its donor pool. Three hygiene rules:

• Convex hull. The treated unit must be spannable by donors. Northgate barely is — that’s the +6.8% offset.
• Plausible donors. Drop markets that are structurally unlike the treated unit; a good numerical fit on irrelevant donors is a mirage.
• No spillover (SUTVA). A donor contaminated by the treatment (e.g. a neighbour that also adopted) is not a clean control. (O’Riordan 2025; Di Stefano-Mellace 2024)

Close the construction act with the practitioner’s discipline that separates a credible synthetic control from a numerical accident. Three rules. First, the convex-hull requirement we just diagnosed: the treated unit must lie inside (or close to) the span of the donors, or no convex blend can match it — Northgate barely qualifies, and that marginal position is the source of its +6.8% offset. Second, plausibility: only include donors that are structurally comparable to the treated unit; a tight numerical pre-fit achieved using irrelevant or implausible donors is overfitting, not matching, and it will not project credibly into the post-period. Third, SUTVA / no spillover: a donor that was itself affected by the treatment — a neighbouring market whose customers also adopted the app, say — is contaminated and biases the counterfactual; donor pools must be clean of treatment and its spillovers. These are active research areas — point students to O’Riordan 2025 on donor selection and Di Stefano and Mellace 2024 on the inferential consequences of hull violations and donor choice, as plain-text pointers, not exam material. The message: weights only mean something if the pool that produced them is hygienic.

Inference

No standard error — so permute the donors

Is +15.2% distinguishable from noise? With one treated market and one realized path, there is no sampling distribution — no classic standard error. Abadie’s answer is a randomization test. (Abadie, Diamond, and Hainmueller 2010, 2015)

Placebo-in-space, three steps:

1. Reassign treatment to each donor in turn (fake launch, week 53); re-run synthetic control → its gap path.
2. Score each unit by its MSPE ratio — post-period ÷ pre-period mean-squared gap.
3. Rank by that ratio: Northgate’s rank ÷ number of units = the Fisher p-value.

Why the ratio, not the raw gap

MSPE = the mean squared gap between a unit and its synthetic. Dividing post ÷ pre asks whether the gap grew after treatment relative to the pre-fit — so a placebo whose synthetic never tracked it can’t top the ranking just by being noisy.

With one treated unit there is no conventional standard error: classical inference needs either repeated draws from a population or many treated units to average over, and synthetic control on Northgate has neither — one treated market, one realized path, no replication. Abadie’s original SC inference replaces the sampling thought-experiment with a RANDOMIZATION (permutation / Fisher) one: reassign treatment to each donor in turn as if IT launched at week 53, re-run the full synthetic-control procedure using the rest as its pool, and score each unit by its MSPE RATIO. Define MSPE plainly: the mean squared prediction error — the average squared gap between a unit and its synthetic over a window. Stress that the RATIO (post-period MSPE ÷ pre-period MSPE), not the raw gap, is the statistic — it normalizes for how well the synthetic fit before treatment, so a unit whose synthetic never tracked it (large pre-MSPE) cannot rank high just from a big, noisy post gap. Rank all units by the ratio; the Fisher p-value is Northgate’s rank divided by the number of units. The ‘terrible pre-fit’ refinement the cache reports is the fit-filter that drops placebos whose pre-MSPE exceeds 5× the treated unit’s. The next slide reads the actual result — deliberately not a slam dunk, and it surprises: Northgate has the biggest gap yet ranks 2nd, because donor M29’s near-perfect pre-fit gives it a higher ratio.

Northgate’s gap stands out — but not overwhelmingly

Northgate’s gap is the biggest, yet it ranks 2 of 16 (Fisher p = 0.125; fit-filtered 2 of 15, p = 0.133). The test scores by MSPE ratio (post ÷ pre), so M29’s near-perfect pre-fit outranks Northgate’s bigger-but-noisier gap — not a larger effect.

Honest reading: with 15 donors and an imperfect fit, single-unit SC inference is fragile. Corroborate across methods — don’t hang the decision on one p-value.

Deliver the honest verdict — this mirrors S10’s deliberately non-triumphant HonestDiD framing and you must not soften it. Northgate’s post-launch gap is by far the BIGGEST in the fan (the eye confirms it), yet it ranks only 2 of 16. Be precise about WHY, because the figure is counter-intuitive: the ranking statistic is the MSPE RATIO — post-period MSPE divided by PRE-period MSPE — not the raw gap. The rank-1 unit is M29 (purple), whose gap is tiny but whose pre-fit is near-perfect (pre-MSPE ~0.0003), so its ratio blows up (8.3); Northgate’s own imperfect pre-fit — the +6.8% hull-edge offset — deflates its ratio (4.9). So M29 outranks Northgate not because its effect is bigger but because the ratio rewards a tight pre-fit. The eye-catching bottom line, M35 (grey dashed at ≈−0.6), is a DIFFERENT unit — a terrible-fit placebo whose synthetic never tracks it; its pre-fit is so bad (pre-MSPE 0.335, ~68× Northgate’s) that Abadie’s fit-filter DROPS it, which is why we also report the filtered rank 2 of 15 (p = 0.133). By the conventional 0.05 bar neither clears significance — with only fifteen donors the permutation floor is about 1/16. The correct conclusion is not “no effect” and not “significant effect”; it is that single-unit synthetic-control inference is FRAGILE — even a visually dominant treated unit can rank 2nd once the pre-fit-sensitive statistic is applied — so do not stake a board decision on one p-value from one method. The constructive move, exactly as in S10, is corroboration: a real prediction interval (next slides), an augmented estimate, and cross-method agreement. The fragility is the argument for the rest of the deck, not a defeat.

Modern SC ships a real prediction interval

The placebo test asks “is it extreme?” Newer theory asks the sharper question: how uncertain is the counterfactual itself? (Cattaneo et al. 2025)

The synthetic counterfactual is \(\widehat{Y}^{(0)}_t=\sum_{j}\widehat{\omega}_j\,Y_{jt}\). Its prediction error has two sources: in-sample — the weights \(\widehat{\omega}\) are estimated on a finite pre-period — and out-of-sample — irreducible noise in predicting a future path.

What scpi reports

A \((1-\alpha)\) prediction interval for the counterfactual \(Y^{(0)}_t\) — bounding the in-sample error and modelling the out-of-sample noise. Read the effect as \(\widehat{\tau}_t=Y^{\text{obs}}_t-\widehat{Y}^{(0)}_t\): the actual above the whole band means \(\tau_t>0\). A band for \(Y(0)\), not a sampling SE for the point.

Introduce scpi as the modern, principled complement to the placebo test, and give the formal object. The placebo test answers a ranking question — is the treated gap extreme among placebos — but it does not quantify how uncertain the counterfactual prediction is. Cattaneo, Feng, Palomba and Titiunik’s scpi framework does. Write the object down: the synthetic counterfactual is the weighted donor combination Ŷ(0)_t = Σ_j ω̂_j Y_jt (the dashed line on the next slide), and its prediction error decomposes into two pieces — the IN-SAMPLE error from estimating the donor weights on a finite pre-period, and the OUT-OF-SAMPLE irreducible noise of predicting a future path. scpi bounds the first (over the weight-constraint set) and models the second (a distributional bound), combining them into a (1−α) PREDICTION INTERVAL for Y(0)_t — the shaded band on the next slide. Stress the interpretation, because students will misread it: this is a band for the COUNTERFACTUAL Y(0), NOT a frequentist standard error on the point. You read the effect as actual minus counterfactual, τ̂_t = Y_obs − Ŷ(0)_t; the actual lying ABOVE the whole band means a positive effect at that confidence level. A conformal-inference alternative (Chernozhukov, Wüthrich, and Zhu 2021) (Chernozhukov, Wüthrich, Zhu) gives a similar prediction guarantee under a permutation-style assumption — a pointer, not a detour. The next slide reads the actual scpi band on Northgate.

Actual above the band → a positive effect

Orange dashed = the synthetic counterfactual \(\widehat{Y}^{(0)}\); orange band = its scpi prediction interval (for \(Y(0)\)); teal = Northgate’s actual path. The gap between them is the effect: the honest effect band is [+6.1%, +10.1%], near the true +8.3% and well below the naive ≈+15% point. The actual sits above the band → a positive effect, and the naive point was overconfident.

Read the figure carefully — this is the act’s payoff. The orange DASHED line is the synthetic counterfactual Ŷ(0) = Σ ω̂_j Y_jt; the shaded ORANGE band is its scpi prediction interval; the TEAL line is Northgate’s actual observed path. The naive classic point estimate was +15.2%, but the honest effect band — the actual minus the counterfactual band — is roughly [+6.1%, +10.1%], centered NEAR the true +8.3% and well BELOW the naive point. Two readings: (1) the actual sits ABOVE the whole band → a positive effect (the band for Y(0) does not reach the actual, so “no effect” is excluded at this level); (2) the band lies below +15.2% → the naive point was OVERCONFIDENT, because it inherited the +6.8% pre-offset. So scpi both confirms a positive effect and corrects the magnitude toward the truth: the interval reveals what the point hid. (The lab reproduces this band in both R scpi and Python scpi_pkg.)

Augmented SC

If you can’t fit inside the hull, correct outside it

Northgate is outside the donors’ hull, so no convex blend reaches it — the +6.8% offset. Augmented SC adds a ridge outcome-model correction that steps outside the simplex just enough. (Ben-Michael, Feller, and Rothstein 2021)

• SCM (classic SC) — on the hull boundary, the closest it can get.
• Ridge ASCM (augmented) — steps outside toward the treated unit.
• Uniform (DiD average) — deep inside; the path runs green (in simplex) → red (in hull) → blue (out of hull).

Source: Ben-Michael, Feller & Rothstein (2021)

Open the augmented-SC act with the fix for the offset, and read the figure. The dashed convex hull is the set of counterfactuals classic SC can build — non-negative weights summing to one. The treated unit (X) lies OUTSIDE it, so no convex blend reaches it; the classic SC solution, SCM, is where the regularization path first touches the hull boundary — the closest feasible synthetic. Follow the coloured path from the Uniform DiD-style average (black square, deep inside): green while the weights stay in the simplex, red while still inside the convex hull, and blue once the path leaves the hull and heads toward the treated unit. Ridge augmented SC (Ridge ASCM, the circled point) sits just OUTSIDE the boundary: the ridge outcome-model correction lets the effective weights step slightly beyond the simplex — small negative or above-one weights — to kill the residual imbalance, with the ridge penalty stopping the extrapolation from running all the way to the treated X. The cleanest analogy, calling back to matching, is bias-corrected matching: match as well as you can, then regress away the leftover imbalance. Source: Ben-Michael, Feller and Rothstein (2021). The next slide writes the estimator down; the slide after shows it working on Northgate (+8.1%).

Augmented SC = outcome model + residuals

Correct the classic weights’ leftover pre-period imbalance with an outcome model \(\widehat{m}(X)\): (Ben-Michael, Feller, and Rothstein 2021)

\[\begin{aligned} \widehat{Y}^{(0),\text{aug}}_{1T} &=\underbrace{\sum_i \widehat{\omega}^{sc}_i\,Y_{iT}}_{\text{classic SC}}+\underbrace{\Big(\widehat{m}(X_1)-\sum_i \widehat{\omega}^{sc}_i\,\widehat{m}(X_i)\Big)}_{\text{imbalance correction}}\\ &=\underbrace{\widehat{m}(X_1)}_{\text{outcome model}}+\underbrace{\sum_i \widehat{\omega}^{sc}_i\big(Y_{iT}-\widehat{m}(X_i)\big)}_{\text{SC-reweighted residuals}} \end{aligned}\]

• Doubly robust: consistent if either the SC weights balance the pre-period or \(\widehat{m}\) is correct.
• Ridge outcome model: \(\widehat{m}(X)=\widehat{\eta}_0+X'\widehat{\eta}_x\), penalty \(\lambda\) by CV — the weights may leave the simplex, the ridge curbs extrapolation.

Write the augmented-SC estimator down — this is the formal core the old deck carried, and it makes “correct the imbalance” precise. Read the top line left to right: the augmented counterfactual is the classic SC prediction PLUS an imbalance-correction term — the outcome model’s prediction for the treated unit minus the SC-weighted outcome-model predictions for the donors. Algebra rearranges this into the second, doubly-robust form: the outcome model’s own prediction for the treated unit PLUS the SC-weighted residuals of the donors (their actual post-outcome minus what the outcome model predicted). That AIPW-like structure is exactly why augmented SC is DOUBLY ROBUST: if the SC weights already balance the pre-period the residual term vanishes in expectation, and if instead the outcome model m̂ is correct the first term carries it — you only need ONE of the two to be right. The outcome model here is a ridge regression of the donors’ post-treatment outcomes on their (centered) pre-treatment outcomes, penalty lambda chosen by cross-validation over pre-periods. The ridge is what lets the effective donor weights leave the simplex — go slightly negative or above one — to reach a treated unit outside the hull, while the penalty stops the extrapolation running wild. Other outcome models are possible (elastic net, random forest, the factor model that becomes gsynth, matrix completion); ridge is the default and the one augsynth uses. The next slide shows the payoff on Northgate: the offset collapses and the estimate snaps to +8.1%.

Augmented SC pulls the overshoot toward truth

Closing the pre-fit (offset ≈ 0.000) collapses the overshoot: augmented SC = +8.1%, 95% CI [+7.6%, +8.6%] — essentially the true +8.3%. Classic +15.2% → augmented +8.1%.

The payoff slide of Act 4 — the bias-correction works almost perfectly, and you can show why. The overlay contrasts the classic counterfactual (red, sitting persistently below real Northgate pre-launch — the +6.8% offset made visible) with the augmented counterfactual (green, lying essentially on top of real Northgate before week 53 — a near-perfect pre-fit, offset approximately 0.000). Because the augmented method drives the pre-period imbalance to zero, the contamination term in the +15.2% = +6.8% + +8.3% decomposition vanishes, and the estimate snaps back to +8.1% with a 95% confidence interval of [+7.6%, +8.6%] — statistically indistinguishable from the true +8.3%. Walk the arc out loud: classic SC +15.2% (inflated by a bad fit) becomes augmented SC +8.1% (a good fit, the truth). This is the mirror of S10’s correction story — there a too-LOW estimate was pulled up toward the truth; here a too-HIGH estimate is pulled down. But do not let students conclude “perfect fit = done”: the very next slide (Ferman) is the discipline that a great pre-fit is necessary, not sufficient.

Fit is a diagnostic, not a verdict

A great pre-fit feels like proof. It isn’t — fit is necessary, never sufficient.

The Ferman caution

Overfitting the pre-period ≠ an unbiased counterfactual. With many donors and a short pre-period you can drive the pre-fit to zero and still be biased — the weights fit noise, not structure. (Ferman and Pinto 2021)

So report three things, never one

1. The fit — pre-period RMSPE / gap plot (necessary, not sufficient).
2. An interval — scpi band or placebo p-value (how uncertain?).
3. Cross-method agreement — do augmented / SDID / DiD concur?

This is the discipline S10 demanded — “corroborate with synthetic control.” The honesty rule: a point estimate without a band is not a finding.

This is the conceptual pivot of the act — the synthetic-control analog of S10’s “flat pre-trend does not prove parallel trends,” plus the reporting rule it forces. Augmented SC just produced a near-perfect pre-fit and the right answer, which is seductive: it tempts you to treat pre-fit quality as proof. Ferman’s caution says no — when the donor pool is large relative to the pre-period, the optimizer can drive the pre-period discrepancy to zero by fitting NOISE rather than structure; such weights fit beautifully but project poorly, so the counterfactual can be biased despite a flawless-looking fit. So a good pre-fit is NECESSARY (a bad fit, like classic Northgate’s +6.8% offset, guarantees contamination) but never SUFFICIENT. The discipline that follows: never report a synthetic-control point alone — report THREE things. One, the FIT (pre-period RMSPE and the gap plot). Two, an INTERVAL (the scpi prediction band or the placebo p-value) to quantify how uncertain the counterfactual is. Three, CROSS-METHOD AGREEMENT — do the augmented, synthetic-DiD, and DiD estimates concur? Convergence across methods with independent failure modes is the real credibility check, exactly as in S10’s robust-estimator forest. Cash the promise S10 made when it ended by telling students to “corroborate with synthetic control”: we have now built that corroboration, and the honesty rule transfers verbatim — a point estimate without a band and a robustness check is not a finding. This is the through-line of all of Part V.

Synthetic DiD

DiD weights time; SC weights units

The two methods we’ve used weight different axes of the same panel:

• DiD weights time: it gives every clean control unit weight 1, and identifies off the common time trend (the period effects).
• Synthetic control weights units: it solves for donor weights \(\omega\), and gives every pre-period equal weight.

Each fixes one axis and optimizes the other. The obvious question: why not weight both?

Open the hinge act by framing DiD and SC as two halves of one design — this is the intellectual climax of the session. Look at the panel as a matrix of units by time. DiD and synthetic control each impute the missing Y(0) by weighting, but on DIFFERENT axes. DiD weights TIME: it implicitly treats every clean control unit equally (weight 1) and does its work through the period fixed effects, which capture the common time trend — so its leverage is the time dimension. Synthetic control weights UNITS: it solves for a bespoke set of donor weights omega but treats every pre-period symmetrically — its leverage is the unit dimension. Each method fixes one axis (uniform) and optimizes the other. Stated this way, the next step is obvious and almost forced: if weighting units helps (SC) and weighting time helps (DiD’s period effects), why not do BOTH at once — solve for unit weights AND time weights together? That is synthetic difference-in-differences, the subject of the next slide, and it is robust precisely because it does not have to rely on either axis alone.

Synthetic DiD weights both axes

Arkhangelsky et al.’s synthetic difference-in-differences: solve for unit weights \(\omega\) and time weights \(\lambda\), then run a weighted two-way DiD — three corners of one family. (Arkhangelsky et al. 2021)

Robust to both failure modes

• \(\omega\) (like SC) matches donors to the treated unit → robust to non-parallel trends.
• \(\lambda\) (new) up-weights pre-periods that resemble the post-period → robust to an imperfect pre-fit. It needs neither a perfect SC fit nor exact parallel trends.

Synthetic DiD inherits the strengths of both parents. Arkhangelsky, Athey, Hirshberg, Imbens and Wager’s estimator solves for TWO weight vectors — unit weights omega (as in synthetic control) and time weights lambda (the new ingredient) — then runs a two-way fixed-effects DiD WEIGHTED by both. The schematic is the unifying image: three weighting grids, units down, time across. DiD is uniform on BOTH axes (every clean control weight 1, every period equal). SC is sparse on the UNIT axis, uniform on TIME. SDID is sparse on BOTH — a few donors AND a few informative pre-periods. So they are three CORNERS of one weighted-two-way-DiD family. The double-robustness callout is the load-bearing payoff: omega makes donors resemble the treated unit, relaxing parallel trends (you need only the matched blend to move in parallel, not the raw average); lambda up-weights the pre-periods that most resemble the post-period, relaxing the perfect-pre-fit demand (residual misfit on irrelevant pre-periods is down-weighted, not leaked). Headline: SDID needs NEITHER a perfect SC pre-fit (which Northgate cannot achieve in the simplex) NOR exact parallel trends (which the omitted regional-economy wave violates) — which is why it lands near the truth while classic SC overshoots. This also seeds the S12 matrix view: once everything is “choose unit and time weights,” the next question is what happens when you model the whole matrix instead.

DiD, SC, and SDID are one weighted regression

Synthetic DiD is a two-way fixed-effects regression with the squared errors reweighted by \(\widehat\omega_i\,\widehat\lambda_t\):

\[(\widehat\tau^{\,sdid},\widehat\mu,\widehat\alpha,\widehat\beta)=\arg\min_{\tau,\mu,\alpha,\beta}\ \sum_{i=1}^{N}\sum_{t=1}^{T}\big(Y_{it}-\mu-\alpha_i-\beta_t-D_{it}\,\tau\big)^2\,\widehat\omega_i\,\widehat\lambda_t\]

\(\mu\) intercept · \(\alpha_i\) unit (market) effects · \(\beta_t\) period (week) effects · \(D_{it}=1\) for the treated market post-launch · \(\tau\) the ATT

Estimator	Unit FE (α)	Weights on the squared errors
DiD	yes	uniform ω, uniform λ (drop out) → plain TWFE
SC	omitted	unit weights ω only; uniform λ
SDID	yes	both ω and λ

All three keep the period effects \(\beta_t\) — DiD’s “time” leverage (the common trend), not a time weight. They differ only in whether the unit FE \(\alpha_i\) is kept and which of the weights \(\omega,\lambda\) are non-uniform.

This is the intellectual spine of the section — the formal version of “three corners.” Read the display equation: SDID is exactly a two-way fixed-effects regression (unit effects alpha_i, time effects beta_t, treatment D_it with coefficient tau = the ATT) whose squared residuals are REWEIGHTED by the product of a unit weight and a time weight, omega_i·lambda_t. The comparison table is the payoff: DiD is the same regression with uniform weights (they cancel) → plain TWFE; SC drops the unit fixed effect and keeps only unit weights (uniform time); SDID keeps the unit+time FE and BOTH weight vectors. So the whole literature is one weighted regression, differing only in (i) whether alpha_i is present and (ii) which weights are non-uniform. Full first-stage math (keep off the face): the unit weights align the donor pre-trajectory to the treated up to a level shift — omega-hat = argmin over a free intercept omega_0 and omega in the simplex of the summed squared pre-period gap between the treated average and omega_0 + the omega-weighted controls, plus a ridge penalty zeta²·T0·||omega||² on the UNIT weights only; the time weights align the pre-periods to the post-period average summed over CONTROL units — lambda-hat = argmin over a free intercept lambda_0 and lambda in the simplex, near-unregularized. The FREE INTERCEPTS are the whole point: they let the weights match the SHAPE of the treated pre-trend, not its level, which is why SDID tolerates a non-parallel trend (omega) and an imperfect pre-fit (lambda). The regularization asymmetry (ridge on omega, none on lambda) reflects that there are many units but few informative pre-periods. RECONCILE THE “DID WEIGHTS TIME” SLIDE: those use “weight” in two senses — the earlier slide means DiD’s leverage is the TIME dimension via the period fixed effects beta_t (present in ALL THREE estimators), whereas here “weights” are the omega·lambda multipliers on the squared errors, which DiD leaves uniform so they factor out (“drop out”). DiD does not up-weight periods; only SDID makes lambda non-uniform. Define mu (overall intercept) and D_it (=1 for the treated market Northgate in its post-launch weeks, 0 otherwise; tau is its coefficient, the ATT) if asked.

SDID lands near the truth on Northgate

Northgate’s unit and time weight vectors — now we know what they buy. Synthetic DiD = +7.9% (true +8.3%); plain DiD = +7.7%. Both near the truth, without a perfect SC fit.

Read what the two weight vectors buy. These are Northgate’s unit and time weight bars for the synthetic-DiD fit, and now students can read what each buys: the unit weights (left) match donors to Northgate, relaxing parallel trends; the time weights (right) emphasize informative pre-periods, relaxing the perfect-fit demand. The payoff: synthetic DiD estimates +7.9% on Northgate against the true +8.3% — essentially right — and even a plain DiD on Northgate gives +7.7%, also near the truth. The striking contrast: classic SC, which insists on a perfect convex fit it cannot achieve, overshoots to +15.2%, whereas SDID, which does NOT require a perfect fit, lands at +7.9%. The lesson students should take: the robustness of SDID comes precisely from not betting everything on one axis. The next slide places this in the forest alongside S10’s estimators and reconciles the two sessions.

Watch SDID stay parallel where classic SC drifts

Same three estimators as trajectories: treated Northgate (dark) vs its synthetic control (cyan); the effect is the arrow after week 53.

• SDID (bottom): the \(\lambda\) ribbon marks the pre-weeks where the two lines are forced parallel → a short, well-anchored arrow (+7.9%).
• Classic SC (middle): the synthetic can’t close the level gap → a longer arrow (the +15.2% overshoot).
• DiD (top): matches the trend, not the level — it leans entirely on parallel trends.

That sparse \(\lambda\) ribbon is the same time-weight vector you saw as bars — now as the pre-window that makes the lines parallel.

The synthdid-package diagnostic on Northgate, one panel per estimator. Each panel plots the treated unit (dark) against the omega-weighted synthetic control (cyan); the estimated effect is the ARROW at the post end, and the shaded ribbon along the bottom is the lambda time weights (sparse — many exactly zero). Precise reading of the arrow: tau-hat is a lambda-WEIGHTED two-way difference — (treated post − lambda-weighted treated pre) − (synthetic post − lambda-weighted synthetic pre) — NOT a raw pre/post level gap; the free intercept omega_0 is why the synthetic can sit parallel-but-offset. Walk the three: DiD (top) — the synthetic is essentially the plain donor average, far below Northgate in LEVEL, so it leans entirely on the parallel shift; SC (middle) — the synthetic tracks the level but cannot close the gap cleanly, so the arrow is long (the +15.2% overshoot); SDID (bottom) — the lambda ribbon anchors the pre-window where the lines are forced parallel, giving a short, well-anchored arrow (+7.9%). This is the picture behind the forest’s numbers on the next slide; cross-reference the ribbon to the time-weight bars on the previous slide rather than re-teaching lambda.

The family converges on ~+8%; classic SC stands alone

Classic SC +15.2% is the lone outlier; augmented +8.1% · SDID +7.9% · DiD +7.7% all cluster on the true +8.3%.

Methods that fail in different ways agreeing is the verdict the single placebo p-value couldn’t give — the honest answer is ~+8% with a band, not fifteen.

The reconciliation forest — the centerpiece and the answer to the placebo act’s fragility, now doing double duty as the section’s closing arc (the old duplicate forest slide is gone). Recall the placebo test could not declare significance (rank 2 of 16, p = 0.125) because single-unit, single-method inference is weak. The forest delivers what one p-value could not: CONVERGENCE. Against the true Northgate +8.3%, three methods with independent failure modes — augmented SC +8.1%, synthetic DiD +7.9%, plain DiD +7.7% — cluster tightly on the truth, while classic SC +15.2% stands alone, exposed as the one that trusted a bad fit (the +6.8% hull offset: +6.8 + +8.3 ≈ +15.2). Same credibility logic as S10’s robust-estimator forest. Reconcile the two sessions: S10’s portfolio robust ATT was +4.9% (the average market); Northgate, the flagship, is larger at +8.3%, and the family recovers it. The classic +15.2% is the Northgate analog of S10’s naive +16.3% — a seductive headline demolished by discipline; the honest answer is ~+8% with a band, not fifteen.

Each method buys an assumption — stacked, they’re one matrix

Family	What it buys	What it costs
Parallel-trends family S10 — DiD, CS, BJS	Additive, transparent, cheap; borrows strength across a whole cohort.	Blind to a unit-specific, time-varying factor; needs a credible cohort and testable pre-trends.
Hull + loadings S11 — SC, augmented SC, SDID	Relaxes parallel trends; builds a unit-specific counterfactual that works at N = 1.	Needs the treated unit inside the donor hull and a non-overfit pre-fit; single-unit inference is weak.

The S12 reveal — planted here

Stack \(\omega\) and \(\lambda\) as rows and you’re completing a low-rank matrix — the same latent factors S10 could not see. S12: DiD, SC, SDID and unconfoundedness become one matrix-completion problem under different regularizers.

The closing slide merges the assumption ledger and the matrix-in-disguise teaser. The ledger is the connective tissue of Part V — reused verbatim in S12 and in the A5 memo — with two rows: the parallel-trends family (S10 — DiD, CS, BJS) buys a cheap, additive, transparent estimator that borrows strength across a cohort but is blind to a unit-specific time-varying factor and needs a credible cohort with testable pre-trends (exactly what N=1 Northgate denies); the hull+loadings family (S11 — SC, augmented SC, SDID) relaxes parallel trends and builds a tailored counterfactual for a single unit but must place the treated unit inside the donor hull, resist overfitting, and lives with weak single-unit inference. The A5 memo asks precisely this: which assumption does your estimator buy, and is it plausible HERE? Then the matrix hinge, delivered as a teaser only: every imputed Y(0) cell is a weighted combination of other cells (what omega and lambda do); stack those weight vectors as rows and you are completing a units-by-time matrix that is approximately LOW-RANK — driven by the same latent regional-economy factor S10 could not see. S12 makes it precise: DiD, SC, SDID and unconfoundedness are one matrix-completion problem under different regularizers, and it adds the third ledger row (low-rank / interactive fixed effects) plus the assumption under all of them — strict exogeneity, no feedback from outcomes back to treatment. Land the section on this S12 hinge.

One treated unit was the easy case

Every estimator here weighted donors to build a counterfactual for one unit at one adoption date. Scale that up and the weighting paradigm answers by multiplying bookkeeping.

Many units — one date

A synthetic control per unit (then average), or one fit to their average? The honest default is partial pooling — a dial between the two.

Staggered dates

Split into adoption cohorts, solve an N = 1 fit per cohort in event time, then aggregate — SC’s Callaway–Sant’Anna analog, with one rule: never borrow from an already-treated unit.

Why S12 exists

This scales by adding bookkeeping — and still can’t represent a unit switching off. Impute the whole \(Y(0)\) matrix instead: matrix completion fills every grey cell from one low-rank structure, absorbing many units, staggered timing, and reversal natively.

The closing pivot — the honest limitation of the whole session, and the setup for S12. Say it plainly: every method today (classic SC, augmented SC, synthetic DiD) built a bespoke counterfactual for ONE treated unit at ONE adoption date by weighting donors. The board’s real portfolios are not like that — many stores, adopting at different times. The point of this slide is that the SC family CAN be scaled, but only by piling on machinery, and that the pile-up is exactly what motivates the matrix view. Keep the taxonomy below in these notes, not on the slide.

MANY UNITS, ONE DATE (block). Three routes: (1) a separate SC per treated unit, then average the unit-level gaps; (2) pool — one SC to the treated group’s average; (3) partial pooling in between. The principled default is partial pooling: Ben-Michael, Feller & Rothstein (2022, JRSS-B 84(2):351–381) show the ATT error bound depends on BOTH the per-unit and the average-of-treated imbalance, so neither endpoint is generally optimal. Software: augsynth::multisynth with nu in [0,1] (nu=0 separate, nu=1 pooled). When many-treated × many-donor makes the per-unit fit non-unique, Abadie & L’Hour (2021, JASA) penalized SC (pensynth) restores a unique per-unit match. Synthetic DiD handles the block case NATIVELY — it is the base case of Arkhangelsky et al. (2021, AER) — and this is where its real inference switches on: the jackknife/bootstrap SEs require N_treated > 1, which is exactly why N=1 Northgate had to fall back to placebo variance. scpi fits the block via scdataMulti with a joint prediction interval.

STAGGERED DATES. The universal trick is block-by-adoption-date decomposition: cut the panel into adoption cohorts, solve an N=1-style SC per cohort aligned in event time (k = t − T_i), then aggregate — the SC-family analog of Callaway–Sant’Anna’s ATT(g,t) then aggregation. augsynth::multisynth(time_cohort=TRUE) and scpi’s staggered UQ (Cattaneo, Feng, Palomba & Titiunik 2025, Review of Economics and Statistics — now published, drop the old “forthcoming”) do this for SC/ASC; scpi’s prediction intervals stay valid even at N_treated=1. Staggered synthetic DiD runs a separate SDID per cohort and weight-averages (Clarke, Pailañir, Athey & Imbens 2024, Stata Journal; command sdid). TWO corrections to avoid mis-teaching it: (a) base R synthdid is BLOCK-ONLY — staggered lives in Stata sdid or a third-party wrapper, not base R; (b) staggered SDID as operationalized uses ONLY never-treated “pure control” units and FAILS with no never-treated unit — it does NOT admit not-yet-treated donors, a genuine divergence from Callaway–Sant’Anna. Otherwise (classic SC, multisynth, scpi) the donor hygiene is the S10 forbidden-comparison rule: a unit’s donors must be never-treated OR not-yet-treated over its own imputation window.

THE HINGE. All of the above scales by ADDING bookkeeping — a donor pool per cohort, event-time realignment per unit, a hygiene test per (unit × event-time) cell, an aggregation-weight choice, an inference method that flips with N_treated — and STILL cannot represent treatment reversal (a unit switching back). Whole-matrix Y(0) imputation dissolves it. Read the family as a factorization arc: classic SC imputes ONE ROW of the potential-outcomes matrix with convex donor weights; ridge ASCM regression-imputes that row with a bias model; SDID doubly-weights units and time; and matrix completion (Athey, Bayati, Doudchenko, Imbens & Khosravi 2021, JASA) / interactive fixed effects (gsynth; fect, Liu, Wang & Xu 2024, AJPS) fill EVERY grey cell at once from one low-rank factor structure. Because the estimand becomes “complete the partially observed Y(0) matrix,” arbitrary patterns — many units, staggered timing, even reversal — are absorbed natively. That is why S12 exists. The slide teaches the pivot; these notes let you field the “how exactly?” question.

References

Abadie, Alberto, Alexis Diamond, and Jens Hainmueller. 2010. “Synthetic Control Methods for Comparative Case Studies: Estimating the Effect of California’s Tobacco Control Program.” Journal of the American Statistical Association 105 (490): 493–505. https://doi.org/10.1198/jasa.2009.ap08746.

———. 2015. “Comparative Politics and the Synthetic Control Method.” American Journal of Political Science 59 (2): 495–510. https://doi.org/10.1111/ajps.12116.

Arkhangelsky, Dmitry, Susan Athey, David A. Hirshberg, Guido W. Imbens, and Stefan Wager. 2021. “Synthetic Difference-in-Differences.” American Economic Review 111 (12): 4088–118. https://doi.org/10.1257/aer.20190159.

Ben-Michael, Eli, Avi Feller, and Jesse Rothstein. 2021. “The Augmented Synthetic Control Method.” Journal of the American Statistical Association 116 (536): 1789–1803. https://doi.org/10.1080/01621459.2021.1929245.

Cattaneo, Matias D., Yingjie Feng, Filippo Palomba, and Rocío Titiunik. 2025. “Scpi: Uncertainty Quantification for Synthetic Control Methods.” Journal of Statistical Software 113 (1): 1–38. https://doi.org/10.18637/jss.v113.i01.

Chernozhukov, Victor, Kaspar Wüthrich, and Yinchu Zhu. 2021. “An Exact and Robust Conformal Inference Method for Counterfactual and Synthetic Controls.” Journal of the American Statistical Association 116 (536): 1849–64. https://doi.org/10.1080/01621459.2021.1920957.

Ferman, Bruno, and Cristine Pinto. 2021. “Synthetic Controls with Imperfect Pretreatment Fit.” Quantitative Economics 12 (4): 1197–1221. https://doi.org/10.3982/QE1596.

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