通过改进的安慰剂检验进行合成控制推断
Inference for Synthetic Controls via Refined Placebo Tests
AI总结：
针对合成控制方法中小样本推断的问题，提出留二法改进安慰剂检验，在均匀/非均匀分配下保持有限样本第一类错误保证，提升小样本分辨率与功效，还支持敏感性分析。
摘要：
合成控制方法常应用于单处理单元和少量控制单元的问题。此场景下常见的推断任务是检验关于处理单元平均处理效应的原假设。渐近合理的推断程序往往不尽如人意，原因包括：（1）小样本量导致大样本近似脆弱；（2）实际执行中估计程序的简化。另一种选择是置换推断，它与一种称为安慰剂检验的常见诊断相关。当处理均匀分配时，置换推断在有限样本中具有可证明的第一类错误保证，且无需简化方法。尽管具有这种稳健性，安慰剂检验仍存在分辨率低的问题，因为原分布仅由N个参考估计构建，其中N是样本量。这为α=0.05等常见水平的统计推断设置了障碍，尤其是当N较小时。我们提出了一种新颖的留二法程序，在广泛的N值下，在均匀分配下仍保持相同的有限样本第一类错误保证，同时绕过了这一问题。与第一类错误始终等于理论上界的安慰剂检验不同，我们的程序通常实现比理论建议更低的无条件第一类错误；这使得在α<1/N的挑战性情况下能够进行有用的推断。从经验上看，当效应量足够大时，我们的程序实现更高的功效，否则功效相当。我们将程序推广到非均匀分配，并展示如何进行敏感性分析。从方法论角度看，我们的程序可被视为一种不同于置换或基于秩的推断的新型随机化推断，在小样本中特别有效。
英文摘要：
The synthetic control method is often applied to problems with one treated unit and a small number of control units. A common inferential task in this setting is to test null hypotheses regarding the average treatment effect on the treated. Inference procedures that are justified asymptotically are often unsatisfactory due to (1) small sample sizes that render large-sample approximation fragile and (2) simplification of the estimation procedure that is implemented in practice. An alternative is permutation inference, which is related to a common diagnostic called the placebo test. It has provable Type-I error guarantees in finite samples without simplification of the method, when the treatment is uniformly assigned. Despite this robustness, the placebo test suffers from low resolution since the null distribution is constructed from only $N$ reference estimates, where $N$ is the sample size. This creates a barrier for statistical inference at a common level like $α= 0.05$, especially when $N$ is small. We propose a novel leave-two-out procedure that bypasses this issue, while still maintaining the same finite-sample Type-I error guarantee under uniform assignment for a wide range of $N$. Unlike the placebo test whose Type-I error always equals the theoretical upper bound, our procedure often achieves a lower unconditional Type-I error than theory suggests; this enables useful inference in the challenging regime when $α< 1/N$. Empirically, our procedure achieves a higher power when the effect size is reasonably large and a comparable power otherwise. We generalize our procedure to non-uniform assignments and show how to conduct sensitivity analysis. From a methodological perspective, our procedure can be viewed as a new type of randomization inference different from permutation or rank-based inference, which is particularly effective in small samples.
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