We propose principled prediction intervals to quantify the uncertainty of a large class of synthetic control predictions or estimators in settings with staggered treatment adoption, offering precise non-asymptotic coverage probability guarantees. From a methodological perspective, we provide a detailed discussion of different causal quantities to be predicted, which we call `causal predictands', allowing for multiple treated units with treatment adoption at possibly different points in time. From a theoretical perspective, our uncertainty quantification methods improve on prior literature by (i) covering a large class of causal predictands in staggered adoption settings, (ii) allowing for synthetic control methods with possibly nonlinear constraints, (iii) proposing scalable robust conic optimization methods and principled data-driven tuning parameter selection, and (iv) offering valid uniform inference across post-treatment periods. We illustrate our methodology with an empirical application studying the effects of economic liberalization in the 1990s on GDP for emerging European countries. Companion general-purpose software packages are provided in Python, R and Stata.

翻译：我们针对交错处理采纳场景，提出系统化的预测区间方法以量化大规模合成控制预测或估计量的不确定性，并提供了精确的非渐近覆盖概率保证。方法学层面，我们详细阐述了待预测的多种因果量（称为"因果预测变量"），可处理处理采纳时间可能不同的多个处理单元。理论层面，我们的不确定性量化方法在以下方面改进现有研究：(i) 覆盖交错采纳情境下的大类因果预测变量，(ii) 支持可能包含非线性约束的合成控制方法，(iii) 提出可扩展的鲁棒锥优化方法与基于数据的系统化调参选择，(iv) 实现处理后时期有效的一致推断。我们通过1990年代经济自由化对欧洲新兴国家GDP影响的实证研究展示该方法论，并提供配套的Python、R与Stata通用软件包。