Difference-in-Differences

The foundational identification assumption of DiD. It posits that, in the absence of the treatment, the average outcomes for the treated and control groups would have followed parallel paths over time. This is inherently untestable for the post-treatment period, so researchers must provide evidence of parallel trends in the pre-treatment period.

• Violation occurs when groups are on divergent trajectories before the intervention.
• Statistical tests like Granger causality are used to check pre-treatment divergence.
• A common robustness check involves adding group-specific linear time trends to the model.

A generalization of DiD that constructs a data-driven counterfactual by creating a weighted combination of untreated units. Unlike DiD, which uses a simple average of a control group, SCM optimizes weights so the synthetic unit closely matches the treated unit's pre-intervention characteristics and outcome trajectory.

• Ideal for comparative case studies with a single treated unit and many controls.
• Eliminates researcher discretion in control group selection.
• Weights are constrained to be non-negative and sum to one, preventing extrapolation.

The canonical regression specification for DiD: Y_it = α_i + γ_t + τ * D_it + ε_it. This model includes fixed effects for each unit (α_i) to absorb time-invariant unobserved heterogeneity, and fixed effects for each time period (γ_t) to control for common macro-level shocks. The coefficient τ on the interaction term D_it estimates the Average Treatment Effect on the Treated (ATT).

• Recent econometric literature warns that TWFE can produce biased estimates with heterogeneous treatment effects and staggered adoption.
• Clustering standard errors at the unit level is essential to account for serial correlation.

An extension of the classic 2x2 DiD design where units receive treatment at different points in time. The simple TWFE estimator can produce severely biased results here because already-treated units serve as controls for later-treated units, creating forbidden comparisons.

• New estimators like Callaway & Sant'Anna and Sun & Abraham solve this by calculating group-time average treatment effects.
• These methods aggregate effects using valid comparisons: never-treated units or not-yet-treated units.
• Essential for analyzing phased technology rollouts or gradual policy implementations across regions.

A dynamic version of DiD that estimates treatment effects for each period relative to the intervention, rather than a single average post-treatment effect. The specification includes leads and lags of the treatment indicator: Σ β_k * D_it^k.

• Leads (pre-treatment coefficients) test the parallel trends assumption; they should be statistically indistinguishable from zero.
• Lags (post-treatment coefficients) reveal the dynamic path of the causal effect—whether it phases in gradually, spikes, or decays.
• Visualized with an event-study plot showing coefficient estimates and confidence intervals over time.

An extension of DiD that adds a third dimension of comparison to address violations of the parallel trends assumption. DDD nets out a second difference—often a comparison between two subgroups within the treated and control groups—to remove confounding trends that affect the treated and control groups differently.

• Example: Comparing a policy change in one state (treated) vs. neighboring states (control), and high-income individuals (treated subgroup) vs. low-income individuals (control subgroup).
• The estimator is the difference in two DiD estimates.
• Requires the less stringent assumption that the bias is the same across the third dimension.