Dube et al. propose the "local projection (LP) based difference-in-differences approach" (LP-DID). In the setting with just two groups and T periods, where all units are treated at the same time, the LD-DID estimator is numerically identical to the "static two-way fixed effects estimator" (TWFE). I find this equality mind-boggling because the TWFE estimator uses all data (all units and all periods) while the LP-DID estimator uses only a sub-sample (it is essentially a cross-sectional regression of the differenced outcome on the treatment group indicator using the sub-sample of all units in the first treatment period). It feels like the TWFE estimator "pretends to use the whole sample but really throws away information." Can someone explain, intuitively and mathematically, why or how the two estimators are numerically identical?
