Fixed Effects Model - Equality of Fixed Effects (FE) and First Differences (FD) Estimators When T=2

Equality of Fixed Effects (FE) and First Differences (FD) Estimators When T=2

For the special two period case, the FE estimator and the FD estimator are numerically equivalent. To see this, establish that the fixed effects estimator is: {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]


Since each can be re-written as, we'll re-write the line as:



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