Validate DID and IV assumptions rigorously

1. Starting from the parallel trends assumption, derive the 2×2 difference-in-differences estimand $\tau = (\bar{Y}_{T,post} - \bar{Y}_{T,pre}) - (\bar{Y}_{C,post} - \bar{Y}_{C,pre})$ and show that it matches the coefficient on a treatment×post interaction in a two-way fixed effects regression when treatment effects are homogeneous. 2) Explain why two-way fixed effects estimators become biased under staggered adoption and heterogeneous treatment effects; compare the Sun–Abraham and Callaway–Sant’Anna estimators and describe how to build an event-study using proper cohort weights. 3) State exactly how you would decide the clustering level for standard errors—and when you would instead use a wild cluster bootstrap—given household-level interference and market-level shocks; discuss the consequences of having too few clusters. 4) Propose a plausibly exogenous instrument for reminder exposure (for example, exogenous send-throttling or an email provider outage that differentially delayed reminders), write the 2SLS setup including the first stage and structural equation, and give the GMM moment conditions. 5) Describe how to test instrument strength and validity—Stock–Yogo weak-IV thresholds, the first-stage F, and the Hansen J overidentification test—under heteroskedasticity and clustering, and interpret a case where $F \approx 8$ and J is insignificant.

Overview: This question tests mastery of causal inference and applied econometrics, including difference-in-differences, two-way fixed effects, staggered adoption and treatment heterogeneity, instrumental variables through 2SLS, clustering, and GMM-based inference.