Before revision:  To address this difficulty we propose two bootstrap methods that can consistently approximate the limit distribution : the gradient bootstrap and the multiplier bootstrap of the inverse propensity score weighted (IPW) estimator. The gradient bootstrap is free of tuning parameters but requires knowledge of the pair identities. The multiplier bootstrap of the IPW estimator does not require such knowledge but involves one tuning parameter. Both methods are straightforward to implement and able to provide pointwise confidence intervals and uniform confidence bands that achieve exact limiting coverage rates. We demonstrate their finite sample performance using simulations and provide an empirical application to a well-known dataset in macroinsurance .
After revision:  To address this difficulty we propose two bootstrap methods that can consistently approximate the limit distribution : the gradient bootstrap and the multiplier bootstrap of the inverse propensity score weighted multiplier bootstrap can be implemented without knowledge of pair identities .