FD is FE’s cousin, but in Stata, reg d.y d.x (manual first-differencing) gives different standard errors than xtreg, fd due to how Stata handles time gaps. For T=2, FD=FE. For T>2, FD is less efficient if errors are serially uncorrelated. But if errors follow a random walk, FD beats FE. Most Stata users never check.
There are three primary foundational models used to analyze static linear panel data. A. Pooled OLS Model stata panel data
gen wage_lag1 = L.wage // previous period gen wage_lead1 = F.wage // next period gen wage_diff = D.wage // change from t-1 to t FD is FE’s cousin, but in Stata, reg d
Allows the intercept ($\alpha_i$) to vary across entities. It "sweeps out" the time-invariant unobserved heterogeneity. FD is FE’s cousin