Anton Strezhnev - Decomposing Triple-Differences Regression with Staggered Adoption

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 (DD) is satisfied. TD designs augment the conventional 2x2 DD with a “placebo” stratum – observations that are nested in the same units and time periods as the DD but are known to be entirely unaffected by the treatment. However, many TD applications go beyond this simple 2x2x2 setting and use observations on many units across multiple time periods and with many “placebo” strata. A popular estimator is the “triple-differences” regression (TDR) fixed-effects estimator – an extension of the common “two-way fixed effects” estimator for DD. This paper decomposes the TDR estimator into its component two-group/two-period/two-strata triple-differences and shows that interpreting this parameter causally in settings with arbitrary staggered adoption requires strong assumptions of homogeneity in the treatment effect not only over time but also across strata. Moreover, under certain forms of treatment staggering, the regression triple-differences estimator no longer consists exclusively of 2x2x2 triple-differences but rather a mixture of triple and double-differences, the latter of which are valid only if parallel trends hold. The decomposition illustrates the importance of being cautious when implementing triple-differences designs in settings more complex than the 2x2x2 case and suggests that alternative approaches such as regression imputation (Borusyak et al., 2021; Liu et al., 2021) may be more appropriate.