Interpret and validate regression with interactions

Consider modeling 7‑day retention (retained_7d, 0/1) using user‑level data with the linear probability model and with logistic regression. Features: treated (1 if exposed to new preloading), watch_time_day1 (minutes), new_user (1/0), and fixed effects for country and signup_date. The model includes an interaction treated × new_user. (a) Write both model specifications (LPM and logit) and interpret the coefficients on treated and on treated × new_user. For logit, translate a coefficient into an odds ratio and then into an approximate marginal effect at the mean. (b) Discuss the assumptions and diagnostics you would check: heteroskedasticity, separation, multicollinearity (e.g., VIF), misspecification (e.g., link test), and calibration (e.g., reliability plots). What standard errors would you report and why (HC-robust vs clustered)? Cluster choice justification. (c) Suppose watch_time_day1 is endogenous (e.g., driven by unobserved preference). Propose two remedies and their assumptions: control function/IV approach and panel fixed‑effects with within‑user variation. What instruments or proxies could be plausible here? (d) You also have count data for daily videos watched. When would Poisson or negative binomial be preferred over OLS? Explain how you’d check overdispersion and interpret the exponentiated coefficients.