Regression discontinuity design (RDD) is widely adopted for causal inference under intervention determined by a continuous variable. While one is interested in treatment effect heterogeneity by subgroups in many applications, RDD typically suffers from small subgroup-wise sample sizes, which makes the estimation results highly instable. To solve this issue, we introduce hierarchical RDD (HRDD), a hierarchical Bayes approach for pursuing treatment effect heterogeneity in RDD. A key feature of HRDD is to employ a pseudo-model based on a loss function to estimate subgroup-level parameters of treatment effects under RDD, and assign a hierarchical prior distribution to ''borrow strength'' from other subgroups. The posterior computation can be easily done by a simple Gibbs sampling, and the optimal bandwidth can be automatically selected by the Hyv\"{a}rinen scores for unnormalized models. We demonstrate the proposed HRDD through simulation and real data analysis, and show that HRDD provides much more stable point and interval estimation than separately applying the standard RDD method to each subgroup.

翻译：回归断点设计（RDD）被广泛用于在由连续变量决定干预的情境下进行因果推断。尽管在许多应用中研究者关注处理效应在子组间的异质性，但RDD通常面临子组样本量较小的问题，导致估计结果极不稳定。为解决此问题，我们提出了分层RDD（HRDD），一种用于探索RDD中处理效应异质性的分层贝叶斯方法。HRDD的一个关键特征是采用基于损失函数的伪模型来估计RDD下处理效应的子组水平参数，并赋予分层先验分布以从其他子组“借力”。后验计算可通过简单的吉布斯采样轻松实现，最优带宽可通过未归一化模型的Hyvärinen评分自动选择。我们通过模拟和实际数据分析展示了所提出的HRDD方法，并证明相较于对每个子组单独应用标准RDD方法，HRDD能提供稳定得多的点估计和区间估计。