  We study estimation and inference on causal parameters under finely
stratified rerandomization designs, which use baseline covariates to match
units into groups (e.g. matched pairs), then rerandomize within-group treatment
assignments until a balance criterion is satisfied. We show that finely
stratified rerandomization does partially linear regression adjustment "by
design," providing nonparametric control over the stratified covariates and
linear control over the rerandomized covariates. We introduce several new
rerandomization schemes, allowing for imbalance metrics based on nonlinear
estimators. We also propose a novel minimax scheme that uses pilot data or
prior information to minimize the computational cost of rerandomization,
subject to a strict bound on statistical efficiency. While the asymptotic
distribution of generalized method of moments (GMM) estimators under stratified
rerandomization is generically non-normal, we show how to restore asymptotic
normality using ex-post linear adjustment tailored to the stratification. This
enables simple asymptotically exact inference on superpopulation parameters, as
well as efficient conservative inference on finite population parameters.
