Causal Inference Workbench DiD · event study · synthetic control · regression discontinuity

y_it = α_i + λ_t + δ·D_it + ε_it, where D_it = 1 once unit i is treated. Both sets of fixed effects are removed by alternating demeaning, δ is estimated by OLS on the residualised variables, and the variance uses the cluster-robust sandwich with clusters at the unit level and the usual small-sample factor G/(G−1)·(n−1)/(n−k). With a single adoption date this equals the 2×2 comparison of means shown in the table. With staggered adoption, TWFE mixes clean and “forbidden” comparisons (Goodman-Bacon 2021); read the event study instead.

Relative time k = t − g_i, where g_i is the adoption period. The plain version regresses y on unit and period effects plus an indicator for every k in the window (endpoints binned), omitting k = −1. The Sun–Abraham (2021) version interacts each cohort with each k, uses never-treated units as the comparison, and averages cohort-specific coefficients with weights equal to each cohort’s share of the treated units observed at that k; its standard errors come from the delta method on the full coefficient covariance. Pre-period coefficients near zero are the parallel-trends check; the simulator’s “steeper pre-trend” box shows what a violation looks like.

τ = lim_x↓c E[y|x] − lim_x↑c E[y|x], estimated by two local linear regressions, one on each side of the cutoff, with triangular kernel weights that fall to zero at the bandwidth h (Hahn, Todd and van der Klaauw 2001). Standard errors are heteroskedasticity-robust (HC1) from the two weighted fits; this is the “conventional” interval, not the bias-corrected one of Calonico, Cattaneo and Titiunik (2014), which the page does not implement. The default bandwidth follows Imbens and Kalyanaraman (2012): a pilot bandwidth gives the density at the cutoff and the conditional variances on each side, cubic and then quadratic fits give the second derivatives, a regularisation term guards against tiny curvature, and the plug-in constant for the triangular kernel is 3.4375. The bandwidth-sensitivity plot re-estimates τ from half to twice that bandwidth. The McCrary (2008) test bins the running variable, fits the density on each side of the cutoff with a local linear regression on the bin heights, and tests whether the log density jumps; a jump means units are sorting across the cutoff and the design is compromised. Placebo cutoffs re-run the estimator at the median of each side, where no effect should appear.

Weights w ≥ 0 with Σw = 1 over never-treated donors, chosen to minimise the pre-treatment root mean squared prediction error between the treated unit and Σ_j w_j·y_jt (pre-period outcomes are the predictors). The problem is solved by accelerated projected gradient descent with an exact projection onto the simplex (Duchi et al. 2008). If several units are treated at the same date, their average is the treated series. Inference is the placebo-in-space test: every donor is treated as if it were the treated unit, and the ratio of post- to pre-period RMSPE for the real treated unit is ranked against the placebo ratios; the rank-based p-value is the share of units with a ratio at least as large.