The Synthetic Control Method

A single exchange changes its tick-size rules and an analyst wants to know the effect on trading volume. There's no other exchange that's a close match on its own, one is too small, another trades a different mix of instruments, a third has different market hours. No individual comparison exchange is a fair control. The synthetic control method's insight: don't pick one imperfect comparison, build a custom blend of several imperfect comparisons, weighted so the blend's pre-change history tracks the treated exchange closely, and let that blend's post-change path stand in for what would have happened without the rule change.

An analogy: mixing a paint color you don't have

You need a specific shade of green paint but the store only sells blue, yellow, and a slightly-off green. Instead of using any one can as an imperfect substitute, you mix a custom blend, 40% blue, 35% yellow, 25% off-green, that matches your target shade as closely as possible when tested on a swatch you can already see (the "pre-treatment" period). Once you're confident the blend matches on the swatch you can check, you trust the same blend recipe to tell you what the color would look like on a wall you haven't painted yet (the "post-treatment" period), the mixing weights, not any single can, are what does the work.

The method, one symbol at a time

Let $Y_{1t}$ be the treated unit's outcome at time $t$, and $Y_{jt}$ the outcome for each of $J$ untreated "donor" units. Choose non-negative weights $w_j$ summing to 1 that minimize the gap between the treated unit and the weighted blend during the pre-treatment period $t < T_0$:

In plain English: find the combination of donor units, using only nonnegative weights that add to one, so it's a genuine weighted average, never an extrapolation beyond the data, that best reproduces the treated unit's own history before the change. The estimated treatment effect at any later time $t \ge T_0$ is simply:

the treated unit's actual outcome minus what the synthetic blend (extended forward using the same weights) would predict.

Worked example 1: constructing the blend

Suppose three donor exchanges have pre-change monthly volumes (in $bn) over three months: Donor A: 10, 11, 12. Donor B: 6, 5, 7. Donor C: 20, 19, 21. The treated exchange: 12.8, 12.7, 13.9. Solving the weighting problem might yield $w_A = 0.5$, $w_B = 0.3$, $w_C = 0.2$ (illustrative, found by minimizing squared pre-period gaps): synthetic path $= 0.5(10,11,12) + 0.3(6,5,7) + 0.2(20,19,21) = (5+1.8+4, 5.5+1.5+3.8, 6+2.1+4.2) = (10.8, 10.8, 12.2)$, reasonably close to but not exactly matching (12.8, 12.7, 13.9); in practice you'd refine weights further, but this illustrates the mechanics: the blend tracks the shape of the treated unit's history using only real donor data.

Worked example 2: reading the post-treatment gap

Suppose after refinement the synthetic blend tracks the treated exchange near-exactly pre-change (gap under 0.1 every month), and the tick-size rule changes in month 4. Post-change: treated exchange volume is 11.5, while the synthetic blend (same weights, extended using donors' actual month-4 volumes) is 13.6. The estimated effect is $\hat\tau_4 = 11.5 - 13.6 = -2.1$, i.e., roughly a $2.1bn reduction in monthly volume attributable to the rule change. Because the pre-period fit was tight, this gap is much more credible than it would be if the synthetic blend had only loosely tracked the treated exchange beforehand, the tightness of the pre-period fit is itself the evidence for (or against) trusting the post-period gap.

What this means in practice

Synthetic control is the standard tool when you have exactly one (or very few) treated units and a panel of untreated candidates, none of which alone is a convincing match, a common situation for exchange-level or country-level policy changes where randomization was never possible and there's no natural single control group. It shares difference-in-differences' reliance on a parallel-trends-style assumption, but instead of assuming it holds for one chosen control, it constructs a control specifically engineered to satisfy it, and the tightness of the pre-period fit is a direct, visible diagnostic of how much to trust the result, unlike diff-in-diff, where a poor implicit match is easy to overlook.