Decomposing Triple-Differences Regression under Staggered Adoption

TL;DR

This paper analyzes the triple-differences regression estimator used in complex causal inference settings, decomposes its components, and highlights the importance of strong assumptions for valid interpretation, proposing alternative methods for more complex scenarios.

Contribution

It provides a detailed decomposition of the TDR estimator, clarifies its assumptions, and recommends alternative approaches for complex staggered adoption settings.

Findings

Decomposition of TDR into simpler triple-differences components

Highlighting the need for strong effect homogeneity assumptions

Demonstration of alternative imputation method in trade policy application

Abstract

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.