The Wild Bootstrap For Few (treated) Clusters

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Fast and wild: Bootstrap inference in Stata using boottest (2018, methodology): “In this context, MacKinnon and Webb (2018) proposed turning to a subcluster bootstrap, in which the bootstrap error terms are clustered more finely than the CRVE.”
What’s trending in difference-in-differences? A synthesis of the recent econometrics literature (2022, unknown): “Simulations have likewise shown that the cluster wild bootstrap may perform poorly in DiD settings with a small number of treated clusters (MacKinnon and Webb, 2018).”
General Equilibrium Effects of Cash Transfers: Experimental Evidence from Kenya (2019, background): “…procedure may perform poorly in cases where most units in treated clusters are treated and there are few clusters, here at most two thirds of households in the most intensely treated clusters were treated, and there are 84 clusters, far above the 15-20 that MacKinnon and Webb (2018) deem adequate.”
Asymptotic theory for clustered samples (2019, unknown): “Since ε and δ are arbitrary we have established (68) and hence (28).”
Asymptotic theory and wild bootstrap inference with clustered errors (2019, methodology): “In MacKinnon and Webb (2018), we nd that Young's procedure tends 2We ran into computational di culties when we attempted to compute these parameters for G1 = 1.”