Difference-in-Differences (DiD)

The foundational identification assumption of DiD. It posits that in the absence of treatment, the average outcome for the treatment group would have evolved identically to the control group. This is not directly testable for the post-treatment period, but researchers validate it by examining pre-treatment trends. If the two groups diverge before the intervention, the estimated treatment effect is likely biased.

The canonical regression specification for DiD. It models the outcome using unit and time fixed effects:

• Unit Fixed Effects: Control for time-invariant unobserved heterogeneity between groups.
• Time Fixed Effects: Control for common macro-level shocks affecting all units. The coefficient on the interaction term between a post-treatment indicator and a treatment group indicator yields the Average Treatment Effect on the Treated (ATT).

An extension that adds a third dimension of comparison to relax the parallel trends assumption. DDD nets out confounding trends by comparing the DiD estimate in a primary market to the DiD estimate in a placebo market unaffected by the treatment. This controls for treatment-group-specific shocks that coincide with the intervention, providing a more robust counterfactual when two-group trends are suspect.

The core identifying assumption of DiD: in the absence of treatment, the average outcomes for the treated and control groups would have followed parallel paths over time. This is fundamentally untestable because the counterfactual is unobserved, but researchers validate it by examining pre-treatment trends. If the two groups exhibit divergent trajectories before the intervention, the DiD estimator is biased. Common diagnostics include plotting pre-period outcomes and conducting placebo tests on pre-treatment periods.

The canonical regression implementation of DiD: Y_it = α_i + γ_t + δ * D_it + ε_it. The model includes unit fixed effects (α_i) to absorb time-invariant group differences and time fixed effects (γ_t) to control for common macro shocks. The coefficient δ captures the DiD estimand. Recent econometric literature by Goodman-Bacon (2021) and Callaway & Sant'Anna (2021) reveals that with staggered treatment adoption, TWFE yields a weighted average of all possible 2x2 comparisons, some with negative weights, potentially biasing the estimate.

An extension of the canonical 2x2 design where units receive treatment at different points in time. This invalidates the standard TWFE estimator because already-treated units serve as controls for later-treated units, creating forbidden comparisons. Modern solutions include:

• Callaway & Sant'Anna (2021): Group-time average treatment effects
• Sun & Abraham (2021): Interaction-weighted estimator
• Gardner (2022): Two-stage DiD approach
• Borusyak et al. (2024): Imputation estimator

A generalization of DiD for single treated unit cases, developed by Abadie & Gardeazabal (2003) and Abadie et al. (2010). Instead of a simple average of control units, it constructs a weighted combination of untreated units that best reproduces the treated unit's pre-intervention trajectory. The weights are constrained to be non-negative and sum to one, preventing extrapolation. This method formalizes the control group selection process and provides a transparent counterfactual when a natural comparison group is unavailable.

A dynamic extension of DiD that estimates treatment effects for each period relative to treatment rather than a single average effect. The specification includes leads and lags of the treatment indicator: Y_it = α_i + γ_t + Σ β_k * D^k_it + ε_it. Pre-treatment coefficients (leads) serve as a falsification test for parallel trends—they should be statistically indistinguishable from zero. Post-treatment coefficients (lags) reveal the dynamic treatment path, showing whether effects emerge gradually, persist, or fade.

An extension that adds a third dimension of comparison to address violations of parallel trends. DDD estimates the differential effect of a treatment across two subgroups within the treated group, using a control group as a baseline. For example, estimating the effect of a minimum wage law on employment in a specific state (treated vs. control), for teenagers (affected group) vs. adults (unaffected group), before and after the law. This differences out unobserved time-varying confounders that affect both subgroups equally.