Definition of The FWL Theorem: The FWL theorem is the following: Given a statistical model y = X₁b₁ + X₂b₂+ e

where:

y is a vector of values of a dependent variable,

the X's are linearly independent matrices of predetermined variables, and

the e's are errors, we could premultiply the equation by M₁=I-X₁(X₁'X₁)^-1X' which projects vectors in the space spanned by X₁ to zero, and run OLS on the resulting equation M₁y = M₁X₂b₂+ M₁e
and (the theorem says) would get exactly the same estimate of b₂ that OLS on the first equation would have given.

This use of premultiplying is used in the derivation of many estimators: notably IV estimators and FE estimators. (Econterms)

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