Fixed effect

Equality of fixed effects and first difference estimators when T=2

For the special two period case (${\displaystyle T=2}$), the fixed effects (FE) estimator and the first difference (FD) estimator are numerically equivalent. This is because the FE estimator effectively "doubles the data set" used in the FD estimator. To see this, establish that the fixed effects estimator is: ${\displaystyle {FE}_{T=2}=\left[(x_{i1}-{\bar {x}}_{i})(x_{i1}-{\bar {x}}_{i})'+(x_{i2}-{\bar {x}}_{i})(x_{i2}-{\bar {x}}_{i})'\right]^{-1}\left[(x_{i1}-{\bar {x}}_{i})(y_{i1}-{\bar {y}}_{i})+(x_{i2}-{\bar {x}}_{i})(y_{i2}-{\bar {y}}_{i})\right]}$