# Statistics: Frisch-Waugh-Lovell Theorem

Consider a regression of a vector of responses, $Y\,\!$, on two sets of predictor variables contained in two matrices $X_1\,\!$ and $X_2\,\!$. The vector of least-squares regression coefficients on $X_1\,\!$ is $\hat{\beta}_1$ where

$Y = X_1 \hat{\beta}_1 + X_2 \hat{\beta}_2 + e$ with $e' [X_1 X_2] = 0 \,$

Let $Q_2 = I - X_2(X'_2X_2)^{-1}X'_2\,\!$ be the matrix of the orthogonal projection onto the orthogonal complement of $\operatorname{span}(X_2)\,\!$.

Then:

$Q_2 Y =Q_2 X_1 \hat{\beta}_1 + Q_2 X_2 \hat{\beta}_2 + Q_2 e= Q_2 X_1 \hat{\beta}_1 + 0 + Q_2 e$

Now $Q_2 e \perp Q_2 X_1$ since $e'Q'_2Q_2 X_1 = e'Q_2 X_1 =e'(I - P_2)X_1=e'X_1 - e'P_2X_1 =0\,\!$ since $e \perp X_1$ and $e \perp X_2$.

Thus, $\hat{\beta}_1$ is the regression coefficient of $Q_2Y\,\!$ on $Q_2 X_1\,\!$ .

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]

$x \perp y$