A geometric representation of the Frisch-Waugh-Lovell theorem

Abstract

Even though the result recently referred to as the "Frisch-Waugh-Lovell theorem" (FWL theorem, henceforth) has been around for a long time, it is relatively recently that it has been widely used by econometricians as a powerful pedagogical tool to express in a simple and intuitive way many results that often rely on tedious and seldom intuitive algebraic steps, which are also notationally cumbersome. Even though a proof of the FWL theorem can be based entirely on standard algebraic results, the main reason of its increasing popularity is its strong geometric appeal. Recent texts and articles provide a mix between algebraic proofs and geometrical illustrations of the theorem, but none of them presents a fully geometrical proof of the result. The goal of this note is very modest: it extends the standard geometrical representations of the theorem to actually prove it based on geometrical arguments, which should, hopefully, provide a richer understanding of the scope of the theorem.