Seven estimators on Brexit, and a placebo test between them (de Brabander et al. 2025)#

Estimators:: Synthetic Difference-in-Differences (SDID) — mlsynth.SDID; Two-Step Synthetic Control — mlsynth.TSSC; Matching and Synthetic Control (MASC) — mlsynth.MASC; Vanilla Synthetic Control (VanillaSC) — mlsynth.VanillaSC
Source:: de Brabander, E., Juodis, A., & Miyazato Szini, G. (2025), “On the Use of Synthetic Difference-in-Differences Approach with(-out) Covariates: The Case Study of Brexit Referendum,” Econometric Reviews.
Replication type:: Path A — the paper’s empirical results on the authors’ own data, both the headline table and the placebo table that judges it.
Status:: All fourteen cells of Table 1 and all twenty-one of Table 7 reproduce, to the precision the paper prints.
Cases:: benchmarks/cases/brabander_brexit_table1.py, benchmarks/cases/brabander_brexit_insample.py

What the paper does#

Born, Müller, Schularick and Sedláček (2019) put the cost of the Brexit referendum at about 2.4 percent of UK GDP by the end of 2018, estimated with a synthetic control built from other OECD economies. This paper asks a question that sits underneath that one: how much of the answer came from the method, and how much from choices that are usually left unstated?

To find out it runs seven estimators on the same panel — quarterly log real GDP for the UK and twenty-three donor countries — and tabulates what each says the UK lost by the end of 2018 and of 2019. The seven are classic synthetic control; demeaned synthetic control, which lets the donors sit at a different level than the treated unit; synthetic difference-in-differences under three different conventions about which quarters the panel carries; MASC, which blends synthetic control with nearest-neighbour matching; and the augmented synthetic control method, which corrects the fit with a ridge regression.

The three SDID rows are the paper’s real subject. They are the same estimator on the same data; they differ only in bookkeeping:

the panel stops one quarter after the referendum, and the effect is reported for a later quarter — so the number being read lies outside the window the weights were fit on;
the panel runs through to the quarter being reported;
the panel is the pre-referendum quarters plus the reported quarter alone.

Nothing in the method says which to pick, and the paper’s point is that the choice is not innocuous.

Table 1: the headline estimates#

Treatment at 2016:Q3, no covariates in the matching. Losses in percent of GDP.

Table 1, 2016:Q3 block — published against mlsynth#
Method	2018:Q4 paper	2018:Q4 mlsynth	2019:Q4 paper	2019:Q4 mlsynth
SC	3.06	3.04	4.20	4.17
DSC	2.98	2.99	4.12	4.12
SDID (i)	2.76	2.77	3.89	3.91
SDID (ii)	2.79	2.80	3.92	3.94
SDID (iii)	2.79	2.80	3.92	3.94
MASC	2.73	2.73	3.83	3.83
ASCM	3.04	3.04	4.19	4.19

The largest deviation is 0.028, on SC at 2019:Q4; eleven of the fourteen cells agree to better than 0.02. The paper prints two decimals, so this reproduces at display precision throughout.

Cases (ii) and (iii) come out identical, in mlsynth and in the published table alike. That is not rounding. With the ridge switched off, the unit weights are a function of the pre-treatment outcomes only, and the time-weight problem lands on the vertex that puts all of its mass on the final pre-treatment quarter — so the two panels hand the weight programs the same information. Case (i) does differ, because its time weights spread a little mass onto two earlier quarters.

Table 7: which of them to believe#

Table 1 says what each estimator claims; it cannot say which claim to trust. The paper’s answer is a placebo exercise. Move the treatment date back to each of twenty pre-referendum quarters (2010:Q1 through 2014:Q4), where nothing happened and the true effect is zero, and score each method by how far its estimate strays from zero. Smaller is better.

Table 7 — published against mlsynth#
Method	RMSE paper	RMSE mlsynth	MAB paper	MAB mlsynth	MedAB paper	MedAB mlsynth
SC	0.0089	0.0086	0.0072	0.0068	0.0055	0.0051
DSC	0.0087	0.0086	0.0070	0.0068	0.0052	0.0051
SDID (i)	0.0067	0.0066	0.0037	0.0037	0.0016	0.0016
SDID (ii)	0.0134	0.0133	0.0111	0.0109	0.0103	0.0102
SDID (iii)	0.0134	0.0133	0.0111	0.0110	0.0107	0.0102
MASC	0.0080	0.0080	0.0062	0.0062	0.0045	0.0045
ASCM	0.0086	0.0086	0.0068	0.0068	0.0051	0.0051

MAB is the mean absolute bias and MedAB the median absolute bias, across the twenty placebo dates. All twenty-one cells reproduce; the largest deviation is 0.00052 and eighteen of the twenty-one land inside 0.00022.

The ranking survives, which is what the table is for. SDID case (i) is the most accurate of the seven by a wide margin, and cases (ii) and (iii) are the least accurate — roughly twice the error of plain SC. Since the three cases are the same estimator, that spread is a statement about bookkeeping.

One caveat the paper itself makes, and worth repeating here: cases (ii) and (iii) are evaluated four quarters past the placebo date while the other five are evaluated one quarter past, so part of their larger error is the longer horizon rather than the convention. Case (i) against case (ii) is the clean comparison, since both fit a panel that extends past the treatment date, and there the gap is still a factor of two.

What it took to match#

Two settings decide every SDID number on this page, and both are the paper’s rather than mlsynth’s defaults.

The first is the penalty. The authors pass zeta.omega = 0 throughout, switching off the ridge on the unit weights that Arkhangelsky et al. (2021) calibrate from the data. mlsynth expresses this with SDIDConfig.zeta; with the default ridge instead, case (ii) reads 2.53 and 3.60 against the published 2.79 and 3.92.

The second is which counterfactual to read. The paper’s estimator subtracts the time-weighted pre-treatment gap,

\[\hat\tau \;=\; y_{1,T} - y_{0,T}'\boldsymbol{\omega} \;-\; \boldsymbol{\lambda}'\bigl(Y_{1,pre} - Y_{0,pre}\boldsymbol{\omega}\bigr),\]

which is mlsynth’s intercept_adjust=True series. Reading the default un-adjusted counterfactual compares a different quantity to the paper’s number and costs about 0.05 percentage points on Table 1 — small enough to look like a minor disagreement rather than the wrong estimand, which is exactly what makes it worth stating.

One discrepancy in the authors’ own package is worth recording. Both scripts set MASC’s cross-validation with a variable named min_value_five, computed as T_pre - 5, which selects five folds. The published cells do not come from that setting; they come from the fold counts stated in the table notes — eighty folds for Table 1, T_pre - 5 folds for Table 7 — which correspond to a smallest fold of six and five respectively. Under the notes’ reading, MASC reproduces exactly (2.73 / 3.83 in Table 1; 0.0080 / 0.0062 / 0.0045 in Table 7); under the scripts’ reading, Table 7’s MASC row comes out at 0.0060 / 0.0041 / 0.0023. The notes are followed here, and the variable name appears to be the slip.

Provenance#

The panel is benchmarks/reference/brabander_sdid_match/brexit_panel.csv, built by reference.R in the same directory from the authors’ brexit_data_raw_eo_nov2018.mat: log real GDP for twenty-four countries over one hundred quarters, with the twelve countries the authors drop for missing data already removed, and the UK marked as treated from 2016:Q3.

Window indices, the three SDID panel constructions, the placebo loop and the evaluation offsets are all read from the authors’ replication scripts — Treatment effects/Treatment effects - 2016Q3 no covariates - no penalty.R and In-sample across periods/In sample across periods - no covariates - no penalty - {SC DSC SDID, MASC AUGSYNTH}.R.

Both cases are deterministic: no resampling, and MASC’s cross-validation grid is exhaustive rather than sampled. Together they run in under ten seconds.

The Monte Carlo: why the demeaned estimators win#

Tables 1 and 7 say what the estimators claim and which of them tracks a known-zero effect best. Neither can say why, because on real data the truth is unobserved. Section 5 answers that by simulating.

The design calibrates the factor model of Abadie, Diamond and Hainmueller (2010) to this same panel,

\[y_{jt} \;=\; \delta_t \;+\; \boldsymbol{\theta}_t' \mathbf{c}_j \;+\; \gamma_t \mu_j \;+\; \varepsilon_{jt},\]

where \(\mathbf{c}_j\) holds unit \(j\)’s six covariate means, \(\delta_t\) and \(\boldsymbol{\theta}_t\) are fit to the real data, and \(\gamma_t \mu_j\) is an interactive fixed effect whose strength \(\gamma\) the study sweeps. The true effect is zero, so an estimate at the final period is the estimation error, and the paper splits that error into the part traceable to covariates, the part traceable to the common factor, and idiosyncratic noise.

The result is about the middle term. As \(\gamma\) rises, SC, MASC and ASCM pick up a bias that grows with it; DSC and SDID barely move, because demeaning absorbs the factor. Measuring the bias at \(\gamma = 0\) and \(\gamma = 1.5\) and taking the difference:

Bias attributable to the common factor#
Method	paper	mlsynth	ratio to SC (paper / mlsynth)
SC	+0.362	+0.425	1.0 / 1.0
DSC	+0.073	+0.090	5.0 / 4.7
SDID	+0.068	+0.092	5.3 / 4.6

The ratio is what the case asserts, not the level and not the slope. A bias level carries a Monte Carlo standard error of roughly 0.08 at the case’s \(M = 100\) – larger than the levels themselves – and even the slopes move by about 0.05 between \(M = 100\) and \(M = 200\). The ratio is stable across both, because numerator and denominator share the same draws, and it is what the paper’s recommendation actually rests on: demeaning cuts the factor-driven bias by a factor of roughly five.

Two checks, not one#

A Monte Carlo aggregate on its own cannot separate a wrong DGP from a wrong estimator; both show up as cells that miss. So the case carries two independent checks.

The first is per-replication cross-validation. reference.R draws 48 panels from the authors’ DGP in R and records synthdid’s own error on each. mlsynth re-runs those same matrices and reproduces every error to within \(5\times 10^{-3}\) – the documented exact-solve versus min.decrease gap, an order of magnitude below the biases being measured. That pins the estimators without reference to the DGP.

The second is the aggregate table above, run on mlsynth’s own port of the DGP (mlsynth.utils.sdid_helpers.simulate). That pins the port without reference to the estimators.

Where the authors’ code and their prose disagree#

Three places, and the code is followed in each, because the code is what produced the printed tables. All three are asserted in mlsynth/tests/test_sdid_simulation.py so the choices cannot be quietly reverted.

The common factor is \(\gamma(1 - 1/t)\) in the script, where the paper writes \(t\gamma/T_0\). The noise is drawn with standard deviation \(\sigma^2\), where the paper describes variance \(\sigma^2\) – and this one is checkable rather than a judgment call, since the published \(\sigma = 0.25\) RMSEs are \(0.0625\) times the \(\sigma = 1\) ones, which a standard deviation of \(0.25\) cannot produce. Finally the treated unit’s loading \(\mu_1\) is the second largest of the draws rather than an independent uniform; that one is a deliberate design choice rather than a slip, since it places the treated unit inside the donors’ convex hull, so the simplex weight problem is feasible without extrapolation.

What the Monte Carlo case leaves out#

The SC(B) all column, which needs the nested predictor-weight search per replication and is the one expensive piece of this design. MASC and ASCM appear in the aggregate check but not the cross-validation: the authors’ masc package requires Gurobi, and augsynth 0.2.0 warns that its ridge tuning is unstable when the unit and pre-period counts are close – 24 and 24 here – and then errors out, so there is no reference to capture. mlsynth’s ridge ASCM is cross-validated against a live augsynth run in Ridge ASCM — Augmented Synthetic Control (Ben-Michael, Feller & Rothstein 2021), on a panel where augsynth does run.

Seven estimators on Brexit, and a placebo test between them (de Brabander et al. 2025)

Contents