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情境。TD设计通过引入"安慰剂"层——即在相同单位与时间段内嵌套、但已知完全不受处理影响的观测值——对传统2×2双重差分进行了扩展。然而，许多TD应用超越了简单的2×2×2结构，使用跨多个时间段的多单位、多"安慰剂"层的观测数据。一种普遍使用的估计量是三重差分回归（TDR）固定效应估计量，它是用于DiD的常见"双向固定效应"估计量的扩展。本文对TDR估计量进行分解，将其拆解为双组/双时期/双层的子组分，并说明在任意交错采用情境下将这一参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会包含处于处理状态下的观测值。该分解阐明了三重差分回归估计量隐含的识别变异来源，建议研究者在处理比2×2×2更复杂的情境时应审慎使用这类估计量。采用仅纳入"纯净安慰剂"的替代方法（如直接插值反事实）可能更为恰当。本文最后通过将"引力模型"应用于评估WTO/GATT对国际贸易影响的实例，展示了该插值估计量的实用性。