Suppose there are $N$ units and $D$ interventions. We aim to learn the average potential outcome associated with every unit-intervention pair, i.e., $N \times D$ causal parameters. While running $N \times D$ experiments is conceivable, it can be expensive or infeasible. This work introduces an experiment design, synthetic A/B testing, and the synthetic interventions (SI) estimator to recover all $N \times D$ causal parameters while observing each unit under at most two interventions, independent of $D$. Under a novel tensor factor model for potential outcomes across units, measurements, and interventions, we establish the identification of each parameter. Further, we show the SI estimator is finite-sample consistent and asymptotically normal. Collectively, these also lead to novel results for panel data settings, particularly for synthetic controls. We empirically validate our experiment design using real e-commerce data from a large-scale A/B test.

翻译：假设存在$N$个单元和$D$种干预措施。我们的目标是学习每个单元-干预对（即$N \times D$个因果参数）的平均潜在结果。虽然理论上可以运行$N \times D$个实验，但实际操作中成本高昂或难以实现。本文引入一种实验设计——合成A/B测试，以及合成干预（SI）估计量，以恢复所有$N \times D$个因果参数，同时确保每个单元最多仅接受两次干预，且该观测次数与$D$无关。基于一种针对跨单元、测量指标和干预措施的潜在结果的新型张量因子模型，我们论证了各参数的可识别性。进一步证明，合成干预估计量在有限样本下具有一致性且渐近正态。这些结论同时为面板数据场景（特别是合成控制法）提供了新成果。我们通过一个大规模A/B测试的真实电商数据对其实验设计进行了实证验证。