Trading-off Bias and Variance in Stratified Experiments and in Staggered Adoption Designs, Under a Boundedness Condition on the Magnitude of the Treatment Effect

Clément de Chaisemartin (1)

1 ECON - Département d'économie (Sciences Po)

I consider estimation of the average treatment effect (ATE), in a population composed of G groups, when one has unbiased and uncorrelated estimators of each group's conditional average treatment effect (CATE). These conditions are met in stratified randomized experiments. I assume that the outcome is homoscedastic, and that each CATE is bounded in absolute value by B standard deviations of the outcome, for some known B. I derive, across all linear combinations of the CATEs' estimators, the estimator of the ATE with the lowest worst-case mean-squared error. This minimax-linear estimator assigns a weight equal to group g's share in the population to the most precisely estimated CATEs, and a weight proportional to one over the estimator's variance to the least precisely estimated CATEs. I also derive the minimax-linear estimator when the CATEs' estimators are positively correlated, a condition that may be met by differences-indifferences estimators in staggered adoption designs.

Mots clés: Bias-variance trade-off; Average treatment effect; Mean-squared error; Minimaxlinear estimator; Bounded normal mean model; Stratified randomized experiments; Differencesin-differences; Staggered adoption designs; Shrinkage

Domaines: Economies et finances

Clément de Chaisemartin. Trading-off Bias and Variance in Stratified Experiments and in Staggered Adoption Designs, Under a Boundedness Condition on the Magnitude of the Treatment Effect. 2022. ⟨hal-03873919⟩