Frisch–Waugh–Lovell theorem

In econometrics, the Frisch–Waugh–Lovell (FWL) theorem is named after the econometricians Ragnar Frisch, Frederick V. Waugh, and Michael C. Lovell.

The Frisch–Waugh–Lovell theorem states that if the regression we are concerned with is expressed in terms of two separate sets of predictor variables:

${\displaystyle Y=X_{1}\beta _{1}+X_{2}\beta _{2}+u}$

where ${\displaystyle X_{1}}$ and ${\displaystyle X_{2}}$ are matrices, ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}}$ are vectors (and ${\displaystyle u}$ is the error term), then the estimate of ${\displaystyle \beta _{2}}$ will be the same as the estimate of it from a modified regression of the form:

${\displaystyle M_{X_{1}}Y=M_{X_{1}}X_{2}\beta _{2}+M_{X_{1}}u,}$

where ${\displaystyle M_{X_{1}}}$ projects onto the orthogonal complement of the image of the projection matrix ${\displaystyle X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}}}$. Equivalently, M_X1 projects onto the orthogonal complement of the column space of X₁. Specifically,

${\displaystyle M_{X_{1}}=I-X_{1}(X_{1}^{\mathsf {T}}X_{1})^{-1}X_{1}^{\mathsf {T}},}$

and this particular orthogonal projection matrix is known as the residual maker matrix or annihilator matrix.

The vector ${\textstyle M_{X_{1}}Y}$ is the vector of residuals from regression of ${\textstyle Y}$ on the columns of ${\textstyle X_{1}}$.

The most relevant consequence of the theorem is that the parameters in ${\textstyle \beta _{2}}$ do not apply to ${\textstyle X_{2}}$ but to ${\textstyle M_{X_{1}}X_{2}}$, that is: the part of ${\textstyle X_{2}}$ uncorrelated with ${\textstyle X_{1}}$. This is the basis for understanding the contribution of each single variable to a multivariate regression (see, for instance, Ch. 13 in ).

The theorem also implies that the secondary regression used for obtaining ${\displaystyle M_{X_{1}}}$ is unnecessary when the predictor variables are uncorrelated: using projection matrices to make the explanatory variables orthogonal to each other will lead to the same results as running the regression with all non-orthogonal explanators included.

Moreover, the standard errors from the partial regression equal those from the full regression.