Frisch–Waugh–Lovell theorem

Remarks

Multiply $y = X_1 \hat\beta_1 + X_2 \hat\beta_2 + e$ by $M_1$. Since $M_1 X_1 = 0$ and $X_1^T e = 0$, this gives $M_1 y = M_1 X_2 \hat\beta_2 + e$. Since $X_2^T e = 0$ and $M_1 e = e$, the vector $e$ is orthogonal to the columns of $M_1 X_2$. So the last equation is the least squares decomposition of $M_1 y$ on $M_1 X_2$. The matrix $X_2^T M_1 X_2$ is invertible because $M_1$ is symmetric and idempotent and $M_1 X_2$ has rank $p_2$ by the rank hypothesis.

Example: take $X_1$ the column of ones. Then $M_1$ subtracts the mean. The theorem says that the slopes of a regression with intercept equal the slopes of the regression of the centred response on the centred regressors.

Going deeper

For a student

The theorem explains the phrase "the effect of $X_2$ holding $X_1$ fixed": a coefficient in a multiple regression is the slope after partialling out the other regressors. It justifies centring data before a fit with intercept. In econometrics it justifies removing fixed effects or trends before estimating the coefficients of interest. It is a purely algebraic fact about projections, so it holds whatever the distribution of the errors.