DiD Parallel Trends & Event Study

Difference-in-Differences (DiD) & Dynamic Event Study Methods estimate causal treatment effects for non-randomized policy interventions, city-level rollouts, or macro-marketing campaigns; Mathematical formulation: estimating the double difference across groups and time: $\hat{\tau}_{\text{DiD}} = (\bar{Y}_{T, \text{post}} - \bar{Y}_{T, \text{pre}}) - (\bar{Y}_{C, \text{post}} - \bar{Y}_{C, \text{pre}})$; estimated via Two-Way Fixed Effects (TWFE) OLS: $Y_{it} = \alpha + \gamma \text{Treat}_i + \lambda \text{Post}_t + \beta (\text{Treat}_i \times \text{Post}_t) + \epsilon_{it}$, where interaction coefficient $\beta$ captures the causal treatment effect; Inviolable Assumption: Parallel Trends Assumption (in the absence of treatment, the average path of the treatment group would follow the control group trajectory); verified empirically via Event Study dynamic coefficients ($t-k$ pre-trends must be statistically indistinguishable from zero).

City-level policy interventions, national promotional campaign evaluations, and macroeconomic feature rollouts.

Isolates genuine policy lift from macroeconomic seasonality and natural market drift that standard pre/post comparisons conflate.

Derive how the OLS interaction coefficient $\beta$ in TWFE regression models algebraically equals the 4-cell difference-in-differences?

How does Event Study regression evaluate dynamic multi-period pre-trends and post-treatment lag effects?

Explain the Goodman-Bacon decomposition showing why classical TWFE produces negative weighting errors under staggered rollout timings?

Explain how Callaway & Sant'Anna (CS-DiD) estimator fixes staggered treatment heterogeneity using not-yet-treated or never-treated controls?

How to remediate non-parallel baseline trends using PSM-DiD or Synthetic Controls?