Uncertainty quantification is a fundamental problem in the analysis and interpretation of synthetic control (SC) methods. We develop conditional prediction intervals in the SC framework, and provide conditions under which these intervals offer finite-sample probability guarantees. Our method allows for covariate adjustment and non-stationary data, among other practically relevant features. The construction begins by noting that the statistical uncertainty of the SC prediction is governed by two distinct sources of randomness: one coming from the construction of the (likely misspecified) SC weights in the pre-treatment period, and the other coming from the unobservable stochastic error in the post-treatment period when the treatment effect is analyzed. Accordingly, our proposed prediction intervals are constructed taking into account both sources of randomness. For implementation, we propose a simulation-based approach along with finite-sample-based probability bound arguments, naturally leading to principled sensitivity analysis methods. We illustrate the numerical performance of our methods using empirical applications and a small simulation study.
翻译:在分析和解释合成控制(SC)方法方面,不确定性的量化是一个根本问题。我们在SC框架内制定有条件的预测间隔,并提供这些间隔提供有限概率保障的条件。我们的方法允许共变调整和非静止数据,以及其他实际相关的特征。构建时首先指出,SC预测的统计不确定性是由两种不同的随机性来源决定的:一种来源是预处理期(可能误判的)SC重量的构造,另一种来源是后处理期分析治疗效果时无法观察的随机性错误。因此,我们提议的预测间隔是考虑到随机性两种来源的。为了实施,我们建议一种基于模拟的方法,连同基于有限抽样的可能性约束参数,自然导致有原则的敏感性分析方法。我们用经验应用和小型模拟研究来说明我们方法的数字表现。