# Statistics: Frisch-Waugh-Lovell Theorem

### From MathWiki

Consider a regression of a vector of responses, , on two sets of predictor variables contained in two matrices and . The vector of least-squares regression coefficients on is where

- with

Let be the matrix of the orthogonal projection onto the orthogonal complement of .

Then:

Now since since and .

Thus, is the regression coefficient of on .

This is the basis of added-variable plots and an early theorem of Econometrics known as the Frisch-Waugh-Lovell theorem. [Thanks to Barry Smith]