Causal Inference and IV: DID, TWFE, Staggered Adoption, Clustering, and 2SLS

Context: You are analyzing the causal effect of a reminder on an outcome using panel data with units (e.g., households) observed over time, nested within markets. Treatment adoption is potentially staggered across cohorts (markets/entities adopt in different periods), and reminder exposure may be partly exogenous due to operational constraints (e.g., throttling or outages).

1) 2×2 DID estimand and TWFE equivalence

• Derive the 2×2 DID estimand: τ = (Ȳ_T,post − Ȳ_T,pre) − (Ȳ_C,post − Ȳ_C,pre) from the parallel trends assumption using potential outcomes.
• Show equivalence to a two-way fixed effects (TWFE) regression with a treatment×post interaction when treatment effects are homogeneous.

2) TWFE with staggered adoption and heterogeneous effects; modern alternatives

• Explain why TWFE is biased with staggered adoption and heterogeneous treatment effects.
• Describe and contrast the Sun–Abraham and Callaway–Sant’Anna estimators.
• Outline how to compute an event-study with proper cohort weights.

3) Standard errors: clustering and bootstrap

• State precisely how to cluster standard errors given household-level interference and market-level shocks.
• When should you use a wild cluster bootstrap, and what are the consequences of too few clusters?

4) Instrumental variables for reminder exposure

• Propose a plausibly exogenous instrument for reminder exposure (e.g., exogenous send-throttling or an email provider outage that differentially delayed reminders).
• Write the 2SLS setup: first stage and structural equation with appropriate fixed effects.
• Provide the GMM moment conditions.

5) Tests for instrument strength and validity

• Describe tests for instrument strength and validity under heteroskedasticity and clustering (Stock–Yogo weak-IV thresholds, first-stage F, Hansen J overidentification).
• Interpret a case where the first-stage F ≈ 8 and the J test is insignificant.