Bootstrap Inference with a Randomly Assigned Regressor: Covariance Filtering and the Limits of Marginal Resampling

Random assignment can make ordinary least squares (OLS) inference insensitive to outcome dependence, yet iid resampling can still fail because assignment and resampling need not remove the same covariance terms. With binary treatment, the iid variance target differs from the sampling variance by exactly minus aggregate cross-unit covariance of treatment effects. Under a Gaussian first-order limit, positive covariance leads to over-rejection and negative covariance to under-rejection. Two designs can generate the same distribution for each observation but require variance corrections of opposite signs, so no marginal-only variance correction is first-order exact for both. For first-order Gaussian inference, only one covariance component---the covariance carried by the randomized-regressor score---must be recovered. Standard dependence estimators on that score restore validity under suitable ordered or grouped dependence conditions. In simulations with ordered data, fixed-bandwidth calibration reduces several large-bandwidth size distortions.