From DiD to SDID

A ladder of synthetic control estimators, and what Brexit cost the UK

2.8%UK GDP shortfall at end-2018 · SDID

0.96of the time weight on one quarter

23donor countries in synthetic Britain

Carlos Mendez

Nagoya University (GSID)

August 2, 2026

The Problem

Act I

There is only one United Kingdom, and it voted to leave

On 23 June 2016 the UK voted to leave the European Union.

To price that decision we need the UK that stayed. It does not exist in any dataset.

Where do we get the counterfactual?

The fundamental problem of causal inference with a single treated unit. No country is a perfect twin of the UK, so we cannot pick a comparison — we have to build one.

Build it out of the countries we do observe

24 OECD economies, quarterly log real GDP, 1995Q1 to 2020Q4.

• 1 treated unit: the United Kingdom
• 23 donors, from Australia to the United States
• 86 pre-treatment quarters to fit against
• 2 evaluation dates: 2018Q4 and 2019Q4

Give each donor a weight. Require the blend to track the real UK for twenty-one years. Read off the gap afterwards.

One line in a crowd

No single donor tracks the UK — which is why we blend them rather than pick one. But notice 2008-09: every line collapses together. That synchronised movement is a common factor, and it is what the demeaned and time-weighted rungs are built to exploit.

The previously published answer was 2.4%

Born, Müller, Schularick and Sedláček (2019) estimated that the referendum had cost the UK about 2.4% of GDP by the end of 2018.

That number came from one estimator with one set of choices.

This deck climbs the whole ladder and asks whether 2.4% survives.

The Ladder

Act II

Every estimator is the same regression

\[\left(\hat{\tau}, \hat{\alpha}, \hat{\beta}\right) = \arg\min \sum_{j} \sum_{t} \left( y_{j,t} - \alpha_j - \beta_t - w_{j,t}\tau \right)^{2} \hat{\omega}_j \hat{\lambda}_t\]

Outcome on a country effect, a quarter effect and a treatment dummy — weighted by \(\hat{\omega}_j\) and \(\hat{\lambda}_t\).

This is the only equation in the deck worth memorising. Everything that follows is a choice of which weights to fix and which to estimate.

Four settings of the same dials

Rung	\(\omega\)	\(\lambda\)	unit effect
DiD	fixed \(1/J\)	fixed	yes
SC	optimised	none	no
DSC	demeaned	fixed	yes
SDID	demeaned	optimised	yes

Two banks of faders. DiD leaves both flat, SC moves one, SDID moves both.

Rung 0 — Difference-in-differences

Every donor counts the same.

w_did <- rep(1/23, 23)
b_did <- mean(Z1 - Z0 %*% w_did)

4.98% at 2018Q4 — nearly double everything else.

Its pre-treatment fit error is 0.0218, four times synthetic control’s.

Parallel trends, failing in plain sight

DiD is fitting a divergence that began in 2013 and calling it Brexit.

Rung 1 — Synthetic control

\[\hat{\omega}^{sc} = \arg\min_{\omega \in \mathbb{W}} \sum_{t=1}^{T_0} \Big( y_{1,t} - \sum_j \omega_j y_{j,t} \Big)^{2}\]

Weights that are non-negative and sum to one: the blend must stay inside the donors’ convex hull.

Synthetic Britain is roughly one-fifth Hungary, one-fifth the US, one-fifth Japan, one-sixth Canada.

3.06% at 2018Q4. Pre-treatment fit error 0.0057.

The rubber band

Pins on a corkboard, a rubber band snapped around them. Everything inside is reachable by a non-negative blend; nothing outside is. If the UK’s pin fell outside, you would need a negative amount of some country.

Rung 2 — Demeaned SC: allow a level gap

SC has no intercept, so it rejects a blend that moves in perfect parallel but sits slightly below.

Demean first, then add the average pre-treatment gap back as a constant.

\[b^{dsc} = \frac{1}{T_0} \sum_{t} \Big( y_{1,t} - \sum_j \hat{\omega}_j y_{j,t} \Big)\]

The offset is tiny, and that is the finding

Here \(b^{dsc} = +0.0024\) — a quarter of one percent. The estimate moves 3.06% → 2.99%.

A small adjustment means the SC fit was already level-balanced. On a dataset where the treated unit sits awkwardly, this term does real work.

Rung 3 — SDID: choose which quarters to trust

DSC averages all 86 pre-treatment quarters equally. 1995 gets as much say as 2016.

The time-weight problem is the unit-weight problem, transposed.

• \(\omega\): which countries, blended, reproduce the UK?
• \(\lambda\): which quarters, blended, reproduce the treatment quarter?

Same solver, transposed argument, and the cross-sectional mean removed instead of the time-series mean.

And the time weights collapse

96% of the weight lands on a single quarter, 2016Q2.

Log GDP is nearly a random walk, so the best blend of past quarters is the most recent one alone. 2.76%.

The Pivot

Act III

Two ways to be wrong

For any weighted counterfactual, the bias splits exactly in two:

\[\text{Bias} = \underbrace{B^{ext}}_{\text{wrong place}} + \underbrace{B^{int}}_{\text{curved function}}\]

Extrapolation bias — the blend’s characteristics do not match the treated unit’s.

Interpolation bias — they do match, but the outcome bends, so the average of outcomes is not the outcome at the average.

Neither name means what you would guess.

The turkey

You are roasting a 3.4 kg bird. The chart lists 3 kg and 4 kg.

Extrapolation bias: you misread the scale and look up 5 kg. Right chart, wrong row.

Interpolation bias: you read both rows correctly and average the times. Right rows — but roasting time bends with weight, so the average of two times is not the time for the average bird.

Two independent errors. Fixing one does nothing for the other.

Only one row is dark in both columns

SC’s unit weights kill extrapolation bias. Matching kills interpolation bias. SDID’s time weights let one estimator target both.

Rungs 4 and 5 — buy the trade-off, or drop the constraint

MASC blends matching and SC, with the mixing weight cross-validated on pre-treatment data. Here it buys 16% matching. → 2.73%

ASCM drops non-negativity and adds a ridge pull toward the SC weights. Eight donors go negative. → 3.04%

Both land inside the range the simpler rungs already covered.

The Verdict

Act IV

Every rung beats 2.4%

2.73% to 3.06% at end-2018; 3.83% to 4.20% at end-2019.

Which rung? A fire drill

Move the treatment date back to twenty quarters when nothing happened. Every estimate is then pure error.

Method	RMSE	Median abs. error
SC	0.0089	0.0055
DSC	0.0087	0.0052
SDID	0.0067	0.0016
MASC	0.0080	0.0045
ASCM	0.0086	0.0051

The SDID family wins on every measure.

But check the exam before trusting the ranking

The published table grades SDID (ii) and (iii) four quarters ahead and everyone else one quarter ahead.

Matched horizons, \(h = 1\):

SDID (i)	SDID (ii)	SDID (iii)
0.0067	0.0066	0.0066

The three variants are indistinguishable. The published ranking among them is an artefact of the horizon, not a property of the estimators.

Covariates make it worse

The dataset ships six covariates. Matching on them:

• reproduces the published 2.43% exactly, via the Born specification
• raises the placebo error for SC (0.0089 → 0.0092) and DSC (0.0087 → 0.0106)

With 86 pre-treatment outcomes already in the matching set, the covariates are redundant (Kaul et al. 2022) and add only noise.

And the optimiser matters more than you would think

The published 3.06% is where Frank–Wolfe happens to be at 10,000 iterations. The true optimum is 3.04%.

What to take away

1. The bias decomposition is the durable idea. The leaderboard is perishable.
2. Fit the ladder, not a rung. Report the cloud.
3. Run a placebo tournament — and check the horizons match.
4. Covariates are not free. With a long pre-period they may be redundant.

And the answer, with its error bars

Brexit cost the UK roughly 3% of GDP by end-2018, 4% by end-2019 — more than previously published.

But the uncertainty is wide. The permutation p-value is 0.042, the smallest this design can produce, and a 95% interval spans roughly 1% to 4.6%.

Worth dwelling on. The point estimate is far better determined than the interval, which is normal with one treated unit and 23 donors. Anyone quoting “Brexit cost 2.8%” without that interval is overstating what this design delivers.

Materials

Post · carlos-mendez.org/post/r_sc_dsc_sdid

Code · analysis.R (18 figures, full replication) · cheatsheet.R (every estimator in 150 lines)

Data · brexit_analysis.csv, loadable straight from GitHub

Replicates de Brabander, Juodis & Miyazato Szini (2025), Econometric Reviews 44(10), 1617–1646.