Inference based on cluster-robust standard errors in linear regression models, using either the Student's t distribution or the wild cluster bootstrap, is known to fail when the number of treated clusters is very small. We propose a family of new procedures calledthe subcluster wild bootstrap, which includes the ordinary wild bootstrap as a limiting case. In the case of pure treatment models, where all observations within clusters are either treated or not, the latter procedure can work remarkably well. The key requirement is that all cluster sizes, regardless of treatment, should be similar. Unfortunately, the analogue of this requirement is not likely to hold for difference-in-differences regressions. Our theoretical results are supported by extensive simulations and an empirical example.

中文翻译：

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少数（已处理）集群的疯狂引导

当处理的簇数非常小时，使用学生的t分布或野生簇引导程序基于线性回归模型中的簇稳健标准误差进行的推理会失败。我们提出了一系列称为子群集野生引导程序的新过程，其中包括普通的野生引导程序作为一种限制情况。在纯处理模型的情况下，对群集中的所有观察结果都进行处理或不进行处理，后一种方法可以非常有效地工作。关键要求是所有簇大小（无论处理如何）都应相似。不幸的是，这种要求的类似物不太可能适用于差异差异回归。我们的理论结果得到了广泛的模拟和一个经验例子的支持。