Before revision:  We derive the limit distribution of the QTE estimator under MPDs, highlighting the difficulties that arise in analytical inference due to parameter tuning. It is shown that both the naive multiplier bootstrap and the naive multiplier bootstrap of the pairs fail to approximate the limit distribution of the QTE estimator under MPDs because they do not preserve the dependence structure within the matched pairs. 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.
After revision:  Standard multiplier bootstrap inference fails to capture the negative dependence of observations within each pair and is therefore conservative. Analytical inference involves estimating multiple functional quantities that require several tuning parameters. Instead, this paper proposes 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.