This paper proposes an asymptotically valid permutation test for heterogeneous treatment effects in the presence of an estimated nuisance parameter. Not accounting for the estimation error of the nuisance parameter results in statistics that depend on the particulars of the data generating process, and the resulting permutation test fails to control the Type 1 error, even asymptotically.

In this paper we consider a permutation test based on a martingale transformation of the empirical process to render an asymptotically pivotal statistic, effectively nullifying the effect associated with the estimation error on the limiting distribution of the statistic. Under weak conditions, we show that the permutation test based on the martingale-transformed statistic results in the asymptotic rejection probability of $\alpha$ in general while retaining the exact control of the test level when testing for the more restrictive sharp null. We also show how our martingale-based permutation test extends to testing whether there exists treatment effect heterogeneity within subgroups defined by observable covariates. Our approach comprises testing the joint null hypothesis that treatment effects are constant within mutually exclusive subgroups while allowing the treatment effects to vary across subgroups.

Monte Carlo simulations show that the permutation test presented here performs well in finite samples, and is comparable to those existing in the literature. To gain further understanding of the test to practical problems, we investigate the gift exchange hypothesis in the context of two field experiments from Gneezy and List (2006). Lastly, we provide the companion RATest R package to facilitate and encourage the application of our test in empirical research.

本文提出了一种渐近有效的置换检验，用于在存在估计的干扰参数的情况下对异质治疗效果进行检验。不考虑干扰参数的估计误差会导致依赖于数据生成过程细节的统计数据，并且由此产生的排列测试无法控制 1 类错误，即使是渐近的。

在本文中，我们考虑基于经验过程的鞅变换的置换检验，以呈现渐近关键的统计量，有效地消除与估计误差相关的对统计量极限分布的影响。在弱条件下，我们表明基于鞅变换统计量的置换检验导致渐近拒绝概率为$\alpha$一般来说，在测试更严格的锐空值时，同时保留对测试级别的精确控制。我们还展示了我们基于鞅的排列测试如何扩展到测试由可观察协变量定义的亚组内是否存在治疗效果异质性。我们的方法包括测试联合无效假设，即治疗效果在相互排斥的亚组内是恒定的，同时允许治疗效果在亚组之间变化。

Monte Carlo 模拟表明，这里介绍的置换测试在有限样本中表现良好，并且与文献中存在的那些测试相当。为了进一步了解实际问题的测试，我们在 Gneezy 和 List (2006) 的两个现场实验的背景下研究了礼物交换假设。最后，我们提供配套的 RATest R包，以促进和鼓励我们的测试在实证研究中的应用。