Synthetic Control

Concept

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Theoretically, Synthetic Control is a comparative causal inference method used to estimate the effect of an intervention by constructing a weighted combination of untreated units that approximates the characteristics and pre-intervention outcomes of the treated unit. The synthetic control acts as the counterfactual representing what would have occurred in the absence of the intervention. The method was developed for policy evaluation and observational studies where a single treated unit or a small number of treated units are available and conventional regression methods are unsuitable.

Mathematically, Synthetic Control estimates non-negative weights assigned to donor units so that the weighted combination minimises differences between the treated unit and the synthetic control before the intervention. The optimisation is performed subject to constraints that the weights sum to one. The estimated treatment effect is the difference between the observed post-intervention outcome of the treated unit and the corresponding outcome of its synthetic control.

In practice, Synthetic Control is implemented using optimisation algorithms that determine donor weights from pre-intervention covariates and outcome trajectories. In health economics it is applied to evaluate national health policies, reimbursement reforms, vaccination programmes, public health interventions, hospital reorganisation, and other system-level changes where randomised experiments are infeasible.

Purpose

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Used to estimate causal effects of interventions in observational settings by constructing a weighted counterfactual from untreated comparison units, supporting policy evaluation and health economic impact assessment.

Mathematical Formulae

Primary Formula

Estimated treatment effect:

�??t = Y?t ? ?w?Y?t

where:

• �??t = estimated treatment effect at time t
• Y?t = observed outcome for the treated unit
• Y?t = outcome for donor unit j
• w? = donor weight

subject to:

w? � 0

?w? = 1

Supporting Formulae

Weight optimisation:

W* = arg min (X? ? X?W)?V(X? ? X?W)

where:

• X? = predictor vector for the treated unit
• X? = predictor matrix for donor units
• W = vector of donor weights
• V = predictor weighting matrix

Pre-intervention Root Mean Squared Prediction Error:

RMSPE = �[(1/T?) ? ?(Y?t ? ?w?Y?t)�]

Related Mathematical Methods

• Comparative Case Study
• Causal Inference
• Difference-in-Differences
• Matching Methods
• Panel Data Analysis
• Propensity Score Methods
• Placebo Testing
• Root Mean Squared Prediction Error

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Example

A health economist evaluates the impact of a national smoking ban on annual hospital admissions.

The treated country is assigned donor weights:

• Country A: 0.45
• Country B: 0.35
• Country C: 0.20

These weights reproduce the treated country's pre-intervention admission trends.

One year after implementation:

Observed admissions = 8,200

Synthetic control admissions = 9,050

Estimated treatment effect:

�? = 8,200 ? 9,050

= ?850 admissions

The analysis estimates that the smoking ban reduced annual hospital admissions by approximately 850 compared with the counterfactual scenario.

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Excel Implementation

Function	Example Formula	Health Economics Application
SUMPRODUCT	=SUMPRODUCT(WeightRange,DonorOutcomeRange)	Calculate the synthetic control outcome as the weighted average of donor units.
Solver	Minimise RMSPE by changing WeightCells	Estimate donor weights subject to non-negative and sum-to-one constraints.
POWER	=POWER(Observed-Synthetic,2)	Calculate squared prediction errors during optimisation.
SQRT	=SQRT(AVERAGE(ErrorRange))	Calculate the pre-intervention RMSPE.
SUM	=SUM(WeightRange)	Verify that donor weights sum to one.

VBA (Optional)

Automate optimisation of donor weights, calculate treatment effects, and generate placebo and sensitivity analyses for synthetic control studies.

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Sources

• Abadie A, Gardeazabal J. The Economic Costs of Conflict: A Case Study of the Basque Country. American Economic Review. 2003.
• Abadie A, Diamond A, Hainmueller J. Synthetic Control Methods for Comparative Case Studies. Journal of the American Statistical Association. 2010.
• Abadie A. Using Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects. Journal of Economic Literature. 2021.
• Cunningham S. Causal Inference: The Mixtape.
• Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
• NICE Health Technology Evaluation Manual.