DifferenceinDifferenceinDifferences

Difference-in-Difference-in-Differences (DDD), also called triple differences, is a quasi-experimental design that extends the standard Difference-in-Differences (DiD) approach by incorporating a second dimension of variation. While DiD compares outcome changes over time between treated and untreated groups, DDD introduces an additional grouping variable (such as region, industry, or policy subtype) and relies on the interaction of three factors: time, treatment, and the third dimension. The goal is to further control for unobserved shocks that vary across both time and the extra dimension.

In practice, DDD relies on a parallel trends assumption that is conditional on the third dimension. That is, in the absence of the treatment, the evolution of outcomes would be the same across all combinations of treatment status and the additional grouping. Empirically, researchers estimate a regression that includes a three-way interaction term. The coefficient on the Post × Treat × Subgroup interaction is interpreted as the DDD treatment effect, representing the differential impact of the policy for the subgroup that experiences both the treatment and a specific third-dimension characteristic.

A typical specification can be written with fixed effects and covariates, where the key term is the three-way interaction. For example, Y_itst = α + β Post_t + γ Treat_i + δ Subgroup_s + θ (Post_t × Treat_i) + φ (Post_t × Subgroup_s) + ψ (Treat_i × Subgroup_s) + κ (Post_t × Treat_i × Subgroup_s) + X'β + ε_itst, with κ as the DDD estimator.

Applications include evaluating policies implemented differently across regions and industries, or programs targeted by multiple criteria. Limitations include data demands, sensitivity to model specification, and the ongoing need for credible parallel-trends behavior within the combined dimensions.