The triple-differences (TD) design is a popular identification strategy for causal effects in settings where researchers do not believe the parallel trends assumption of conventional difference-in-differences (DiD) is satisfied. TD designs augment the conventional 2x2 DiD with a "placebo" stratum -- observations that are nested in the same units and time periods but are known to be entirely unaffected by the treatment. However, many TD applications go beyond this simple 2x2x2 and use observations on many units in many "placebo" strata across multiple time periods. A popular estimator for this setting is the triple-differences regression (TDR) fixed-effects estimator -- an extension of the common "two-way fixed effects" estimator for DiD. This paper decomposes the TDR estimator into its component two-group/two-period/two-strata triple-differences and illustrates how interpreting this parameter causally in settings with arbitrary staggered adoption requires strong effect homogeneity assumptions as many placebo DiDs incorporate observations under treatment. The decomposition clarifies the implied identifying variation behind the triple-differences regression estimator and suggests researchers should be cautious when implementing these estimators in settings more complex than the 2x2x2 case. Alternative approaches that only incorporate "clean placebos" such as direct imputation of the counterfactual may be more appropriate. The paper concludes by demonstrating the utility of this imputation estimator in an application of the "gravity model" to the estimation of the effect of the WTO/GATT on international trade.

翻译：三重差分（TD）设计是一种流行的因果效应识别策略，适用于研究者认为传统双重差分（DiD）的平行趋势假设不成立的场景。TD设计通过引入"安慰剂"层——即嵌套在相同单位和时间段内但已知完全不受处理影响的观测值——来扩展传统的2x2 DiD。然而，许多TD应用超越了简单的2x2x2结构，使用跨多个时间段、涉及多个单位及多个"安慰剂"层的观测数据。针对此类场景的常用估计量是三重差分回归（TDR）固定效应估计量——这是DiD中常见的"双向固定效应"估计量的扩展。本文将TDR估计量分解为其构成的两组/两期/两层三重差分成分，并说明在任意交错采用的情境下，若要将该参数解释为因果效应，需要强效应同质性假设，因为许多安慰剂DiD包含了处理状态下的观测值。该分解阐明了三重差分回归估计量背后的隐含识别变异，表明研究者在使用此类估计量处理比2x2x2更复杂的情境时应保持谨慎。采用仅纳入"纯净安慰剂"的替代方法（如直接插补反事实）可能更为合适。本文最后通过应用"引力模型"估计WTO/GATT对国际贸易的影响，展示了这种插补估计量的实用性。