Bias-Aware Confidence Intervals for Synthetic Control via Placebo-in-Time Bootstrap

Synthetic control (SC) methods are among the most widely used tools for causal inference without randomization.

The standard Gaussian confidence interval around the estimated effect is simple, fast, and reliably directional when the treatment signal is strong, so practitioners default to it for good reason.

Most treated populations, however, are bottom-heavy in intensity, and for them the SC model's systematic bias rivals or exceeds the signal even under good pre-treatment fit.

Because this bias shares sign across units it does not average out, and the Gaussian confidence interval shrinks past it and converges on a wrong center.

The failure is not imprecision but misdirection: a positive effect estimated as negligible is a missed opportunity, while a negligible effect estimated as significantly positive leads to continued investment in an intervention that is not working.

No existing confidence interval for the SC effect measures this bias.

We propose a placebo-in-time bootstrap that estimates the bias distribution directly from the observed panel.

For each treated unit the procedure backdates the treatment onset and refits the SC model at each placebo onset; the resulting placebo gaps are draws from the same bias distribution that contaminates the real estimate, and bootstrapping them yields a critical value calibrated at the zero null.

Because the method resamples realized model error rather than a hypothesized effect, coverage is trajectory-agnostic: it holds at fixed width regardless of how the true effect evolves over time.